UC-NRLF 


QC 


NOTES 


ON  THE 


PROPERTIES  OF  MATTER  AND  HEAT 


BY 

PERCIVAL    LEWIS,  Ph.  D. 
*» 

Associate  Professor  of  Physics  in  the  University  of  California 

57 


(  UNIVERSITY  ) 


BERKELEY,  CAL. 
1903 


Copyright,  1903,  by  PERCIVAL  LEWIS 


SAN   FRANCISCO,  CAL. 

PRESS   OF  THE   HICKS-JUDD   CO, 

1903 


PREFACE. 

The  following  Lecture  Notes  have  been  privately  printed  for  the  use 
of  students  taking  sophomore  Physics  in  the  University  of  California. 

Students  cannot  be  too  often  reminded  that  they  cannot  really  learn 
the  principles  of  Physics  unless  they  become  acquainted  with  them  at  first 
hand  from  daily  experience  or  from  individual  or  lecture  experiments; 
text-books  are  merely  guides  to  the  use  of  such  material.  They  are 
very  necessary  guides,  however,  for  without  them  it  would  be  impossible 
for  any  one  person  to  properly  classify,  coordinate,  and  interpret  his 
knowledge,  to  fill  the  gaps  in  his  experience,  or  to  get  a  proper  perspective 
view  of  the  whole  subject.  In  reading,  however,  do  not  in  any  case  try 
to  recall  the  statements  as  given  in  the  book.  See,  if  you  can,  the 
thing  or  the  phenomenon  to  which  your  attention  is  called  by  it,  and  try 
to  remember  what  your  own  eyes  have  seen;  if  that  is  impossible,  try  to 
form  a  mental  picture  of  the  thing  or  the  phenomenon,  not  of  the  words 
of  the  book.  A  proper  use  of  the  imagination  in  this  way  will  make  the 
study  a  much  easier  one. 

Some  attention  paid  to  the  historical  development  of  Physics  and  to 
the  biographies  of  the  great  physicists  will  do  much  to  humanize  the  sub- 
ject and  thus  arouse  interest  and  stimulate  the  imagination  and  memory;  it 
will  also  add  greatly  to  the  culture  value  of  Physics. 

In  studying  Physics  one  learns  many  interesting  facts,  some  of  which 
may  be  of  practical  use.  Its  principal  value,  however,  should  be  not  to 
impart  information,  but  to  exercise  the  mind  in  drawing  logical  conclu- 
sions, and  to  form  the  habit  of  seeking  truth  for  its  own  sake  in  fields 
where  traditional  authority  and  personal  prejudice  have  no  weight,  this 
making  it  easier  for  us  to  seek  truth  in  fields  where  these  have  weight. 

Berkeley,  August,  1903. 


114431 


Below  are  the  titles  of  some  good  available  works  of  reference: 

General  Physics. — Text-books  of  Deschanel,  Daniell,  Hastings  and 
Beach,  and  Watson. 

Properties  of  Matter. — Tait,  Properties  of  Matter;  Mach,  Science  of 
Mechanics;  Poynting  and  Thompson,  Properties  of  Matter;  Risteen, 
Molecules;  Boys,  Soap  Bubbles;  Perry,  Spinning  Tops.  The  following 
little  books  of  the  Scientific  Memoir  Series  give  the  original  papers,  with 
bibliography;  Mackenzie,  The  Laws  of  Gravitation ;  Bams,  The  Laws  of 
Gases;  Ames,  The  Free  Expansion  of  Gases;  Jones,  The  Modern  Theory 
of  Solution. 

Heat. — Text-books  by  Maxwell,  Balfour  Stewart,  Tait,  Madan,  and 
Preston,  and  Tyndall's  Heat  as  a  Mode  of  Motion;  Stewart,  The  Conser- 
vation of  Energy. 

Historical. — Whewell,  The  History  of  the  Inductive  Sciences; 
Routledge,  Popular  History  of  Science;  Cajori,  History  of  Physics; 
Clerke,  History  of  Astronomy;  Grant,  History  of  Physical  Astronomy; 
Williams,  The  Story  of  Nineteenth  Century  Science. 

Biographical. — Garnett,  Heroes  of  Science  (for  Boyle,  Cavendish, 
Rumford,  and  Maxwell);  Lodge,  Pioneers  of  Science  (for  Newton  and 
Kepler);  biographical  articles  in  Encyclopedia  Brittanica  and  other 
encyclopedias. 

Such  scientific  journals  as  Science,  Nature,  and  the  Popular  Science 
Monthly  should  be  frequently  consulted. 

Special  references  will  also  be  given  in  the  text. 


CONTENTS. 


Page. 
PROPERTIES  OF  MATTER. 

Introductory 1 

FUNDAMENTAL  MAGNITUDES 4 

Physical  Quantities 4 

Time 5 

Length 5 

Mass 5 

GENERAL  PROPERTIES  OF  MATTER 5 

Extension 6 

Inertia 6 

MOTION 6 

Speed 6 

Angular  Speed 7 

Velocity 7 

Acceleration 7 


FORCE. 


MEASUREMENT  OF  MASS 8 

Density 8 

FORCE  AND  MOTION 9 

Composition  of  Forces 10 

Resolution  of  Forces 10 

Moment  of  Force 11 

Parallel  Forces 11 

Momentum 11 

GRAVITATION 12 

Acceleration  of. 12 

Center  of  Gravity 12 

The  Balance 13 

Kepler's  Laws 14 

Universal  Gravitation 15 

Centripetal  Acceleration 15 

Newton's  Proof. 16 

Direct  Verification 17 

Variations  of. 18 

Tides 19 

Determination  of  g 21 

Falling  Bodies 21 

Inclined  Fall 22 

Nature  of  Gravitation 22 

CONSERVATION  OF  MASS 23 

WORK  AND  ENERGY 23 

SPECIAL  PROPERTIES  OF  MATTER 25 

Elements  25 

Discontinuity 26 

Divisibility 26 

Molecular  Forces ^6 

States  of  Matter ,.  26 


Page. 

FLUIDS 27 

Hydrostatics 28 

Transmission  of  Pressure 28 

Pressure  Due  to  Weight 28 

Pascal's  Principle 28 

Archimedes'  Principle 28 

Specific  Gravity 29 

GASES 30 

Weight  of. 30 

Boyle's  Law 31 

Charles'  Law 31 

Deviations  from  Laws 32 

Amagat's  Results 32 

VanderWaals'  Equation 33 

Dalton's  Law 33 

THE  ATMOSPHERE 33 

Pressure  of. 33 

Barometers 34 

Measurement  of  Altitudes 35 

Cyclones 36 

Manometers 36 

Air  Pumps 37 

Mercury  Pumps 37 

GASES  IN  MOTION 38 

Efflux  of 38 

Transpiration 39 

Pressure  in 39 

Diffusion,  Free 40 

Diffusion,  Partitions 41 

ABSORPTION  OF  GASES 41 

OCCLUSION 41 

ADSORPTION 42 

VISCOSITY  OF  GASES 43 

KINETIC  THEORY  OF  MATTER 43 

PROPERTIES  OF  LIQUIDS 47 

Hydrostatics 47 

Hydrodynamics 47 

Jets 48 

Flow  in  Pipes 48 

Vortices 48 

MOLECULAR  FORCES  IN  LIQUIDS 49 

Compressibility 49 

Cohesion  and  Adhesion 50 

Viscosity 50 

Surface  Tension 50 


CONTENTS— Continued. 


Page. 

SOLUTIONS 54 

Surface  Tension  of. 54 

Solubility 54 

Diffusion 55 

Osmosis 55 

Theory  of. 56 

SOLIDS  -..- 57 

Flow  of 57 

Diffusion  of 57 

Solution  of. 58 

Elasticity 58 

Hardness 59 

Tenacity 60 

Friction 60 

Crystals 61 


MOLECULAR  THEORIES. 


SIZE  OF  MOLECULES 


HEAT. 


Production  of 

Effects  of 

Nature  of. 

Historical 

Quantity  of  Heat 


Gl 


(52 


TEMPERATURE 65 

Scales  of 65 

Thermometers 65 

CALORIMETRY 67 

Quantity  of  Heat 67 

Specific  Heat 67 

Methods  of  Measurement 68 

Specific  Heat  of  Gases 69 

Change  of  Specific  Heat ; 70 

Latent  Heat ....  ,.  70 


MECHANICAL  EQUIVALENT  OF  HEAT 


71 


Page. 

EXPANSION 72 

Expansion  of  Solids 72 

Anomalous  Expansion 73 

Expansion  of  Liquids 74 

Expansion  of  Gases , 75 

Absolute  Zero 76 

CHANGE  OF  STATE 77 

Internal  Work 77 

Fusion 77 

SOLUTION  AND  CRYSTALLIZATION 79 

VAPORIZATION  AND  CONDENSATION 80 

Spheroidal  State 82 

Pressure  of  Saturated  Vapors 83 

Fogs,  Cloud,  Rain,  Dew 84 

Vapor  Density 85 

ISOTHERMAL  CURVES 85 

CONTINUITY  OF  LIQUID  AND  GASEOUS  STATE    86 
LIQUEFACTION  OF  GASES 87 

OTHER  EFFECTS 88 

Molecular  Constraints 88 

Chemical  Action 89 

MEASUREMENT    OF    HIGH     TEMPERATURE 
AND  SMALL  DIFFERENCES  OF 89 

TRANSFER  OF  HEAT 90 

Radiation 90 

Convection 90 

Conduction 90 

THERMODYNAMICS 92 

Carnot  Reversible  Cycle 93 

Absolute  Temperature 93 

Change  of  Freezing  Point 93 

Adiabatic  Expansion 94 

ORIGIN  AND  MAINTENANCE  OF  SUN'S  HEAT    95 
DISSIPATION  AND  DEGRADATION  OF  ENERGY    95 


Vwi*ivt.rv<2ti 

PROPERTIES  OF  MATTER. 


INTRODUCTORY. 

1 .  All  the  natural  phenomena  which  appeal  to  our  senses  are  part  of 
the   subject-matter    of  Physics.       This   is   true    not  only  of  the  simpler 
processes  of  inorganic  life,   but,   to   a   greater   or  less  extent,   of  all  the 
phenomena  of  organic  life,  such  as  the  rise  of  sap,  the  circulation  of  the 
blood,  and  muscular  movements.       Physics  is,   therefore,  the  foundation 
of  all  the  sciences. 

The  most  striking  discoveries  in  Physics  were  made  in  the  last  two  or 
three  centuries,  and  as  a  systematic  science  it  is  scarcely  a  century  old  ; 
yet  it  had  its  unconscious  beginnings  before  any  other  science,  when  man 
began  to  apply  his  mind  to  the  utilization  of  natural  phenomena.  In 
supplying  their  daily  wants  our  distant  ancestors  must  have  mastered  and 
applied  many  rudimentary  principles  of  Physics,  just  as  people  to-day  who 
have  never  studied  Physics  are  constantly  making  use  of  its  principles 
under  the  name  of  common  sense. 

2.  This  knowledge  cannot  properly  be  called  scientific  knowledge, 
however,  for  it  does  not  involve  deliberate  analysis  into  their  simplest  terms 
of  the  phenomena  of  nature,  and  comparison  of  the  results  with  a  view  to 
finding  general  elementary  principles  which  will  give  us  a  clearer  insight 
into  their  nature  and  relations,  and  guide  us  to  further  knowledge.     Those 
who  seek  knowledge  for  immediate   practical  ends  usually  have  neither 
the  time  nor  the  disposition  to  put  it  on  a  scientific  basis,  and  the  present 
state  of  our  knowledge  and  civilization  owes  little  to  such  persons.     The 
progress  of  the  sciences,  and  incidentally  of  the  practical  arts  based  upon 
their  principles,   has  always   been   due   most   largely  to  the  craving  for 
knowledge  for  its  own  sake,   regardless  of  the  material  advantages  which 
it  may  secure,  or  even  with  a  deliberate  sacrifice  of  advantage  or  pleasure ; 
Galileo's  reward  for  truth-seeking  was  imprisonment,  and  Faraday  resigned 
opportunities  of  making  a  fortune  by  applied  science  in  order  that  he 
might  give  all  his  time  to  research.       The  justification  of  such  a  course, 
from  a   material   standpoint,    is    that   practically   all    the   applications  of 
electricity  in  use  to-day  are  based  upon  Faraday's  discoveries  and  would 
be  impossible  without  them. 

Because  science  has  its  origin  in  the  love  of  truth  and  not  in  money- 
getting,  we  find  its  first  beginnings  in  those  directions  where  absolutely  no 
material  reward  could  be  expected — in  the  study  of  the  heavenly  bodies. 
Some  of  the  ancients  were  content  to  observe  and  even  worship  the  sun 
and  other  heavenly  bodies  without  asking  what  their  nature  might  be,  or 
the  cause  of  their  orderly  motions ;  but  there  were  some  who  sought  to 
secure  at  least  enough  information  about  these  bodies  to  correctly  describe 
their  paths  and  predict  their  future  positions.  The  first  physical  laws  to 
be  established  were  those  which  describe  the  motions  of  the  members  of 
the  solar  system. 

3.  Although  we  learn  every  science   through    our  preceptions,  the 
most  important    results   are    usually  not    immediately  perceptible  to  our 
•senses.     We  do  not  see  directly  the   real   motions   of  the  planets  with 


2  PROPERTIES    OF    MATTER. 

reference  to  the  sun,  because  those  motions  are  compounded  with  that  of 
the  earth,  and  so  it  took  many  centuries  of  patient  observation  to  pass 
from  the  epicycles  of  Ptolemy  to  the  ellipses  of  Kepler.  We  cannot 
see  the  materials  in  the  sun  and  the  fixed  stars,  but  the  spectroscope 
enables  us  to  determine  some  at  least  of  their  constituents  with  as  much 
certainty  as  though  they  could  be  analyzed  in  a  chemical  laboratory.  It 
is  not  the  substance  itself,  but  the  quality  of  light  which  it  emits,  which 
appeals  directly  to  our  senses ;  yet  the  proof  is  as  valid  as  the  recognition 
of  a  friend,  which  is  made  possible  by  the  light  he  reflects. 

Direct  or  absolute  knowledge  of  things  is  inconceivable.  Our  knowl- 
edge is  all  through  our  senses,  or  from  inferences  based  on  sense 
perceptions.  We  learn  the  form  and  color  of  certain  things  ;  we  feel  that 
they  are  heavy  or  light,  hard  or  soft,  and  that  they  move  in  such  and  such 
a  way.  Most  people  have  seen  round  things  and  square  things,  red  things 
and  green  things  ; .  they  have  felt  hard  things  and  soft  things,  smelt  things 
that  were  pleasant  or  unpleasant,  have  heard  loud  or  faint  sounds  If  we 
attempt  to  describe  any  new  object  to  them,  k  is  by  the  use  of  such  terms 
which  suggest  known  comparisons.  It  would  evidently  be  impossible  to 
describe  a  red  apple  to  one  who  had  never  seen  the  color  red  or  a  sphere. 
It  may  be  concluded  from  such  considerations  that  all  our  knowledge  is 
relative,  or  based  upon  comparison  with  other  objects  familiar  to  our 
senses. 

4.  Sometimes  we  find  that  many  bodies  differing  widely  in  many 
respects  have  a  common  attribute ;  for  example,  all  bodies  known  to  us 
will  fall  to  the  earth  if  unsupported,  or  have  weight ;  all  kinds  of  matter 
have  the  property  of  inertia,  or  the  inability  to  alter  their  own  motion  in 
any  way.  Such  facts  are  physical  principles,  the  beginnings  of  a  science. 
We  find,  further,  that  certain  phenomena  tend  to  occur  in  a  definite  way; 
that,  for  example,  the  planets  move"  in  approximately  elliptical  orbits  about 
the  sun.  The  statement  of  this  fact  is  a  physical  law.  This  fact  was 
"explained"  later  by  Newton,  who  showed  that  this  type  of  motion 
resulted  from  the  more  general  law  that  every  two  bodies  in  the  universe 
attract  each  other  with  a  force  varying  inversely  as  the  square  of  the 
distance  between  them.  All  so-called  "explanations"  of  physical  facts 
are  of  this  character  ;  the  explanation  consists  in  showing  more  general 
relations  or  a  law  of  wider  scope.  A  law  itself  is  simply  a  description 
reduced  to  its  simplest  and  most  general  terms.  The  reason  why  two 
bodies  attract  each  other  is  unknown  to  us  and  must  always  remain 
unknown.  We  may,  however,  at  some  future  time  analyze  this  attrac- 
tion and  show  that  it  depends  in  some  way  on  a  medium  between  the  two 
bodies.  We  may  then  say  that  we  have  explained  the  attraction  ;  but  the 
explanation  will  simply  reduce  the  attraction  of  distant  bodies  to  a  case  of 
push  or  pull  between  two  contiguous  bodies — and  who  can  explain  the 
why  of  the  push  or  the  pull  ? 

The  student  of  Physics  cannot,  then,  hope  to  find  out  why  things 
happen;  he  can  only  expect  to  find  out  how  they  happen  and  to  formulate 
this  experience  in  general  descriptions  or  laws.  Such  laws  greatly 
economize  mental  effort  by  systematizing  our  knowledge.  The  term  law 
used  in  this  way  has  none  of  the  legal  sense  of  obligation  or  necessity. 

5.  Nature  does  not  always  afford  us  the  best  opportunities  for 
observing  her  phenomena.  They  may  occur  so  rarely  or  so  fitfully, 
or  on  such  a  small  scale,  that  we  cannot  study  them;  or  they  may  be  so 
complex  as  to  make  it  very  difficult  to  resolve  them  into  their  simplest 


INTRODUCTORY.  3 

elements.  We  therefore  resort  to  experimentation,  in  which  we  may 
arbitrarily  bring  about  the  necessary  conditions  and  exclude  those  which 
might  be  confusing.  If,  for  example,  we  wish  to  find  out  how  changes 
of  temperature  affect  a  gas,  we  separate  all  foreign  substances  from  it  and 
arrange  the  conditions  so  that  the  temperature  alone  changes,  all  other 
physical  conditions  remaining  constant. 

6.  The  knowledge  which  we  gain  by  observation  and  experiment  is 
of  use  to  us  chiefly  because  we  believe  that  it  will  continue  to  hold  good 
in  the  future.     It  is  not  merely  a  mass  of  historical  material  describing 
what  has  been;  it  enables  us  to  describe  what  is  going  on  to-day,  and  to 
predict  what  will  take  place  to-morrow  with  a  feeling  of  great  certainty. 
Experience  invariably  teaches  the  lesson  that  the  processes  of  nature  are 
uniform,    or    that    under    the    same    conditions    the    same    routine    of 
phenomena  will  continue.      Because  the  sun  has  risen  and  set  at  regular 
intervals  for  all  recorded  time,  we  believe  that  it  will  continue  to  do  so 
unless   some    catastrophe   occurs   which   alters    the    conditions.       Most 
physical  laws  are  based  upon  the  observation  of  a  large  number  of  cases, 
but  sometimes  a  surprisingly  small  number  will  give  us  a  firm  conviction 
of  the  generality  of  a  law.     We  are  content  to  believe  after  one  experi- 
ment that  all  red-hot  objects  will  burn  us.      Having  once  observed  that  a 
pure  metal  melts  at  a  certain  temperature,  we  believe  that  all  specimens  of 
that   metal  will   do  the  same,    under    the   same    conditions.      It  is    very 
important,  however,  to  be  sure  that  all  conditions  are  the  same.     In  the 
last  case,  for  example,  the  melting  point  would  not  be  the  same  if  the 
pressure  to  which  the  metal  is  subjected  were  changed.     This  process  of 
basing  a  general  law  on  a  limited  number  of   observed  cases  is  called 
scientific  induction.      In  no  case  can  it  be  said  to  give  us  a  certainty,  but 
it  does  give  a  high  degree  of  probability. 

7.  Sometimes   the   progress   of   science   is   greatly    aided    by    the 
judicious  use  of  the  imagination,   constantly  checked    by   a    comparison 
with  facts.     The  closest  observation  will  not  show  us  the  paths  described 
about  the  sun  by  the  planets,  but  we  may,  as  Kepler  did,   venture  the 
hypothesis  that  the  shape  of  an  orbit  is   that  of  an  ellipse,  or  parabola, 
or  some  higher  curve;  and  then  we  can  check  our  guess  by  comparison 
with  given  positions  of  the  planet  and  prove  or  disprove  the  hypothesis. 
Hypotheses   which   cannot    be   tested   and   verified   have   no    claims   to 
consideration. 

In  the  same  way  a  theory,  due  wholly  to  the  imagination,  may  be  of 
the  highest  use.  In  no  way  can  an  atom  or  molecule,  if  such  actually 
exist,  appeal  directly  to  our  senses,  yet  the  molecular  theory  of  matter 
has  been  an  exceedingly  fruitful  one.  We  try  to  picture  to  ourselves 
such  a  constitution  of  matter  that  its  various  physical  properties,  such  as 
elasticity  and  changes  of  state  with  temperature  and  such  phenomena  as 
diffusion  of  gases,  the  pressure  of  gases,  and  chemical  phenomena  would 
be  natural  consequences.  So  long  as  the  molecular  or  any  other  theory 
is  consistent  with  all  known  facts,  and  even  serves  to  predict  new  facts,  it 
matters  little  whether  it  is  literally  true  or  not;  it  is  just  as  good  as  if  it 
were  true,  and  is  probably  as  near  the  truth  as  we  can  ever  get.  If  a 
theory  is  a  sound  one,  it  must  always  be  in  accord  with  the  facts;  hence 
it  is  absurd  to  speak  of  theory  as  being  opposed  to  practice.  If  it  is 
theoretically  true  that  a  given  gas  will  diffuse  through  an  opening  of  a 
given  size  at  a  certain  rate,  we  shall  find  it  to  be  practically  true  if  the 
conditions  are  in  accord  with  those  demanded  by  the  theory. 


4  PROPERTIES    OF    MATTER. 

8.  The  student  must  bear  constantly  in  mind  that  the  matter  in  a  text- 
book is  not  Physics.  The  direct  object  of  his  study  is  nature  as  we  find 
it  in  everyday  life,  supplemented  by  the  laboratory.  Every  one  of  us 
began  his  study  of  Physics  before  he  could  read.  The  text-book  is  a 
guide  to  the  use  of  this  material,  just  as  a  catalogue  is  a  guide  through  an 
art  gallery;  each  is  almost  indispensable  on  account  of  its  classification  of 
details,  its  historical  information,  and  its  discussion  and  criticisms;  but 
neither  can  take  the  place  of  the  original  subject  matter. 


References. — Mach,  Popular  Scientific  Lectures — The  Economical  Nature  of 
Physical  Inquiry — The  Principle  of  Comparison  in  Physics.  Clifford,  Lectures, 
Vol.  I — Aims  and  Instruments  of  Scientific  Thought.  Tyndall,  Fragments  of 
Science — Scientific  Use  of  the  Imagination. 

FUNDAMENTAL    MAGNITUDES    AND    UNITS. 

9.  The   impressions    which    we    receive-  from  the   external    world 
through  our  physical  senses  can  all  be  reduced  to  descriptions  in  terms  of 
what  we  call  time,   space,    and  matter.     Physics  does  not  undertake  to 
discuss  the  ultimate  nature  of  these  fundamental  physical  magnitudes,  but 
leaves  such  questions,  as  well  as  that  of  the  objective  reality  of  matter  and 
phenomena,  to  be  settled,  if  possible,  by  philosophy.    The  ideas  conveyed 
by  the  words  space  and  time  are  so  rudimentary  that  they  admit  of  no 
further  description  or  definition  than  that  furnished  by  ordinary  experi- 
ence.      The  same  may  be  said  of  matter;  but  the  varied  properties  of 
the    latter,    together   with  the  time  and  space  relations  involved  in  its 
varied  activities,  furnish  an  inexhaustible  storehouse  of  material  for  all  the 
physical   sciences,    such   as    Chemistry,    Astronomy,    Geology,    and  the 
applied  sciences.     All  of  these  lie  in   part  at  least  within  the  domain  of 
Physics;  but  in  order  to  limit  the  scope  of  the  science  to  manageable 
dimensions  it  is  usually  understood  to  deal  onlv  with  the  more  general  and 
fundamental  properties  of  matter  and  the  phenomena  which  are  common 
to  all  these  sciences. 

There  are  various  physical  phenomena,  such  as  those  described  under 
the  headings  of  Light  and  Electricity  and  Magnetism,  which,  although 
manifested  in  connection  with  matter,  do  not  directly  involve  its  properties. 
It  will  be  found  as  we  study  these  phenomena  that  they  may  be  consistently 
described  if  we  assume  that  they  take  place  in  an  hypothetical  medium 
called  the  ether,  which  possesses  some  of  the  properties  of  matter,  or  may 
even  be  the  basis  of  all  matter.  It  is  usually  assumed  that  the  ether  does 
not  directly  affect  our  senses,  but  it  may  at  least  be  questioned  whether  the 
action  of  a  dazzling  beam  of  sunlight  upon  the  eye  is  not  as  direct  an  effect 
as  the  blow  of  a  club  upon  the  head.  In  studying  the  phenomena  of  light 
and  electricity  we  shall  find  that  the  reasons  for  assuming  the  existence  of 
the  ether  are  about  as  convincing  as  those  for  the  existence  of  matter. 

10.  Physical  Quantities. — Any  description  of  the  phenomena  of 
nature  which  undertakes  to  be  precise  must  be  founded  on  accurate  meas- 
urements,  requiring  the  use  of  certain  standards  of  comparison,   called 
units.     In  the  absence  of  absolute   knowledge  regarding  what  we  call 
matter,  force,  energy,  heat,  electricity,  etc.,  we  can  measure  them  only  in 
terms  of  some  of  their  invariable  properties  or  effects.     The  term  quan- 
tity has,   therefore,  a  somewhat  arbitrary  meaning,   and  it  might  easily 
happen  that  quantity  measured  in  one  way  might  not  be  consistent  with 


A 


PHYSICAL    QUANTITIES.  O 

that  measured  in  another.  Such,  for  example,  is  the  case  in  measuring 
temperature  by  thermometers  of  different  materials.  The  most  funda- 
mental physical  quantities  are  those  of  time,  length,  and  mass. 

Time. — The  natural  unit  of  time  is  the  solar  day,  or  the  interval 
between  two  successive  transits  of  the  sun  over  the  same  meridian;  this, 
however,  varies  in  length  according  to  the  position  of  the  earth  in  its  orbit. 
The  average  length  of  the  day  during  a  whole  year,  or  the  mean  solar 
day,  is  adopted  as  a  standard.  A  more  convenient  unit  for  ordinary  pur- 
. poses  is  the  second,  of  which  there  are  86,400  in  a  mean  solar  day. 

The  tidal  waves  on  opposite  sides  of  the  earth  are  held  in  a  nearly 
fixed  position  relative  to  the  moon,  while  the  earth  revolves  between  them 
as  a  car  wheel  revolves  against  the  shoe  of  its  brake;  undoubtedly,  there- 
fore, the  speed  of  rotation  of  the  earth  is  diminishing  on  account  of  the 
tidal  friction,  but  the  effect  is  so  small  that  during  historical  time  it  has 
not  reached  a  measurable  amount,  as  shown  by  the  comparison  of  the 
times  of  the  recorded  with  the  calculated  eclipses  of  the  moon  in  ancient 
times. 

Length. — Various  arbitrary  standards  of  length  have  been  adopted  in 
different  countries.  We  need  concern  ourselves  only  with  the  English 
foot  and  yard,  and  the  French  meter,  standards  of  which  are  kept  by  those 
governments.  The  meter  was  intended  to  be  the  one  ten-millionth  part  of 
the  distance  from  the  earth's  equator  to  its  poles,  but  actually  differs 
slightly  from  this. 

1  meter  =  10  decimeters  =  100  centimeters  =  100&  millimeters. 

1  inch  =  2  53995  cm. 

The  meter  was  defined  by  a  law  of  France  in  1795  as  the  distance  between 
the  ends  of  a  platinum  rod  made  by  Borda,  at  0°  C. 

Mass  is  sometimes  defined  as  quantity  of  matter ;  more  exactly  it  is 
proportional  to  the  inertia  or  to  the  weight  of  matter,  these  two  proper- 
ties varying  together  at  any  given  place.  The  English  unit  of  mass  is  the 
pound,  the  French  is  the  kilogram,  which  is  defined  as  the  mass  of  a  cer- 
tain piece  of  platinum  made  by  Borda  in  1795.  The  kilogram  is  intended 
to  be  equal  to  the  mass  of  a  liter  or  1000  cubic  cm.  of  water  at  4°  C.  The 
usual  scientific  unit  of  mass  is  the  gram,  or  the  mass  of  a  cubic  cm.  of 
water  at  4°  C. 

1  kilogram  —  1000  grams  =  2.2046  pounds. 

1  gram  =  10  decigrams  =  100  centigrams  =  1000  milligrams. 
Usually  the  centimeter,   gram,  and  second  are  used  as  scientific  units. 
This  is  called  the  C.  G.  S.  system. 

The  student  should,  by  reference  to  some  standard  text-book  or  labor- 
atory manual,  familiarize  himself  with  the  principles  underlying  some  of 
the  ordinary  instruments  for  precise  measurement,  such  as  the  following: 
Time — Clock,  chronograph.  Length — Vernier,  comparator  or  dividing 
engine,  cathetometer,  micrometer  screw,  spherometer.  Mass — Chemical 
balance. 

GENERAL   PROPERTIES   OF    MATTER. 

11.  No  definition  of  matter  can  add  anything  to  the  knowledge  of  it 
which  we  have  all  gained  by  experience.  All  kinds  of  matter  have  been 
found  to  possess  the  universal  properties  of  extension  and  inertia,  as  well 
as  the  relative  property  of  weight,  so  that  matter  has  been  defined  as  that 
which  possesses  these  properties. 


6  PROPERTIES    OF    MATTER. 

12.  Extension. — We  cannot  think  of  any  material  body  except  as 
occupying  a  definite  region  of  space,  from  which  all  other  bodies  are  ex- 
cluded, and  we  assume  that  no  two  particles  of  matter,  however  small,  can 
simultaneously  occupy  the   same   space.     This   conclusion   is   of  course 
purely  hypothetical.     So  far  as  ordinary  sensible  masses  are  concerned, 
any  apparent  violations  of  the  principle  of  impenetrability  can  always  be 
explained  as  a  result  of  the  porosity  or  non-continuity  of  matter.   Examples 
of  such  apparent  penetrability  are  found  in  the  case  of  solution,  as  of  sugar 
in  water.     Carbon-monoxide  passes  readily  through  red-hot  iron;  Francis  t 
Bacon  forced  water  through  the  walls  of  a  leaden  globe;  an  alloy  of  tin 
and  copper  is  smaller  by   7  or  8  per  cent  than  the  sum  of  the  volumes  of 
its  original  constituents;  the  same  is  true  of  a  solution  of  alcohol  in  water. 
Faraday  found  that  potassium  oxide  has  a  smaller  volume  than  that  of  the 
original  potassium  before  oxidization.     Glass  and  other  vitreous  bodies 
alone   show  no  evidence  of  porosity;  such  Substances  are  not  leaky  to 
anything  known  to  us. 

13.  Inertia. — When  a  car  stops  suddenly  its  occupants  are  pitched 
forward.     A  ball  thrown  vertically  upward  from  the  deck  of  a  moving 
vessel  will  return  to  the  hands  of  the  thrower.       A  pendulum  vibrating 
without  constraint  in  a  given  plane  will  continue  to  vibrate  in  that  plane, 
while  the  earth  rotates  beneath  it;  to  the  observer,  who  is  at  rest  relatively 
to  the  earth,  the  plane  of  the  pendulum  seems  to  shift.     This  is  known  as 
the  Foucault  pendulum.     A  ball  rolling  on  a  horizontal  plane  will  go 
further  and  further *in  a  straight  line  before  it  comes  to  rest  as  the  friction 
'and  other  constraints  are  diminished.     It  is  a  legitimate  conclusion  that  if 
all  resistance  were  removed,  the  ball  would  continue  to  move  forever  in  a 
straight  line.     This  is  the  case  of  ideal  limits,    such  as  often  occurs  in 
mathematical  problems.     A  stone  whirled  in  a  sling  cannot  move  in  a 
straight  line,  because  the  constraint  of  the  cord  holds  it  at  a  fixed  distance 
from  the  center  of  rotation;  but  it  will  fly  off  in  a  straight  line  if  the  string 
breaks.     So  long  as  it  is  held  by  the  string,  it  moves  as  nearly  as  possible 
in  a  straight  line — that  is,  in  a  plane — unless  it  is  violently  displaced.     For 
the  same  reason  a  heavy  wheel  (gyroscope)  offers  a  strong  resistance  to 
any  attempt  to  change  its  plane  of  rotation.     A  bullet  set  in  rotation  by  a 
rifled  barrel  will  go  straighter  to  its  mark  than  one  from  a  smooth-bore 
gun.^1)     We  learn  from  such  facts  that  constancy  of  motion  is  charac- 
teristic of  all  matter  if  left  to  itself;   it  is  said  to   be  inert,    or  to  possess 
inertia,    meaning  thereby   that  it  has  no  power  to  alter  its  own  motion, 
either  in  speed  or  direction.     To  bring  about  such  changes,  some  external 
agency  is  necessary,    and  while  the  change  is  taking  place,  the  external 
agency  is  said  to  exert  a  force  upon  the  body. 

14.  Motion  is  purely  a  relative  term.     All  bodies  known  to  us  are 
apparently  in  motion.     In  order  to  determine  the  displacement  of  any 
object  we  must  consider  some  point — say  a  point  on  the  earth's  surface — 
to  be  at  rest,  and  measure  all  displacements  from  that  point  as/an  origin. 

Jw4^L^AilMMM,  w^^V^ 

Speed  is  rate  of  change  of  o2aUe*i  measured  along  the  Jibe  of  displace- 
ment. If  uniform,  it  is  equal  to  the  displacement  per  unit  time;  if  it 
varies  uniformly,  it  is  equal  to  an  infinitesimal  displacement  divided  by 
the  infinitesimal  time  required  for  it.  If  times  and  displacements  be 
plotted  on  coordinate  axes,  it  will  be  seen  that  in  all  cases  the  speed  is 


(1)  See  Perry,  Spinning  Tops. 


MOTION  —  FORCE. 


proportional  to  the  tangent  of  the  angle  with  the  X  axis  of  the  element  of 
the  curve  corresponding  to  the  instant  at  which  the  speed  is  required. 
In  general,  if  d  is  displacement  along  a  path,  straight  or  curved, 


where  the  interval  t2  —  t^  must  be  taken  infinitesimally  small  if  the  speed 
is  changing. 

Angular  Speed  is  the  rate  at  which  a  line  rotating  about  a  point 
(radius  vector)  is  changing  its  direction.  The  simplest  way  of  reckoning 
angles  is  in  radians.  The  radian  is  an  arc  equal  in  length  to  the  radius 
of  the  circle  described  by  any  point  on  the  radius  vector,  and  in  a  circum- 
ference there  are  2?r  radians.  If  w  be  the  angular  speed, 

-« 


the  latter  term  being  used  if  the  speed  is  uniform.     T  is  then  the  time  of  a 
complete  rotation. 

Velocity  is  speed  in  a  given  direction;  the  term  implies  direction  as 
well  as  rate  of  displacement.  A  speed  in  a  given  direction  may  be 
resolved  into  velocities  along  the  X  and  the  Y  axes,  or  in  any  other 
direction. 

Acceleration  is  rate  of  change  of  velocity,  and  like  velocity,  implies 
direction  as  well  as  change  of  speed.     If  the  acceleration  a  is  uniform, 


If  variable,  the  interval  /2  —  ^  must  be  made  very  small.  The  accel- 
eration is  proportional  to  the  tangent  of  the  angle  with  the  X  axis  of  an 
element  of  the  curve  formed  by  plotting  successive  values  of  time  and 
velocity  along  coordinate  axes. 

Angular  Acceleration  is  defined  in  a  similar  way. 

15.  Force.  —  When  the  motion  of  a  body  is  accelerated,  that  is, 
when  its  motion  is  changed  either  in  speed  or  direction,  or  both,  experi- 
ence shows  that  it  is  due  to  the  action  of  some  external  agency,  and  to 
this  action  we  apply  the  name  force.  It  is  to  be  noted  that  force  is  that 
which  is  either  actually  producing  an  acceleration,  or  which  is  only 
prevented  from  doing  so  by  the  action  of  an  equal  and  oppositely-directed 
force.  Force  does  not  have  the  objective  reality  of  matter  and  of  energy, 
but  may  exist  at  one  instant  and  be  destroyed  the  next.  Press  your 
hand  on  a  table  and  a  force  exists  which  would  move  the  table  were  it  free 
to  move;  take  your  hand  away,  and  the  force  ceases  to  exist. 

It  is  possible  in  several  ways  to  make  quantitative  measurements  of 
force.  For  example,  we  know  that  it  requires  a  certain  force  to  elongate 
a  spiral  spring  by  a  given  amount;  the  free  end  of  the  spring  will  move 
until  the  elastic  force  of  restitution  is  just  equal  to  the  applied  force.  We 
infer  logically  that  it  will  require  twice  the  force  to  elongate  two  parallel 
springs  to  the  same  extent.  Experiment  shows  that  the  force  which  will 
elongate  two  similar  springs  simultaneously  by  a  given  amount  will 
elongate  one  spring  twice  the  amount.  By  such  experiments  we  arrive 
at  the  law,  which  experiment  shows  to  be  very  nearly  true,  that  the 


8  PROPERTIES    OF    MATTER. 

elongation  or  compression  of  a  spring  is  proportional  to  the  force  acting 
upon  it,  and  we  can  use  such  a  spring  in  the  comparison  of  forces.  This 
is  an  example  of  the  statical  method  of  comparison,  in  which  equilibrium 
is  secured  by  balancing  two  equal  forces  against  each  other,  so  that  there 
is  no  resulting  motion. 

If,  however,  we  consider  the  simpler  case  of  a  single  unbalanced 
force,  we  must  measure  it  in  terms  of  the  acceleration  produced  by  it. 
The  force  is  defined  as  being  proportional  to  the  acceleration.  It  is  found, 
however,  that  the  acceleration  does  not  depend  solely  upon  the  force, 
but  also  upon  the  nature  and  size  of  the  body  acted  upon.  That  function 
of  the  nature  and  the  size  of  the  body  which  determines  the  acceleration 
produced  by  a  given  force,  or  in  other  words  which  determines  the 
inertia  of  the  body,  we  call  mass.  Usually,  but  rather  incorrectly,  mass 
is  defined  as  quantity  of  matter.  It  is  doubtful,  however,  whether  the 
term  quantity  of  matter  conveys  any  clear  klea  when  we  are  comparing 
different  kinds  of  matter. 

16.  The  Measurement  of  Mass. — If  mass  be  defined  as  the  quantity 
of  matter,  it  is  easy  to  see  that  two  cubic  centimeters  of  water  must  have 
twice  the  mass  of  one  cubic  centimeter;  it  is  not  at  all  evident  that  a 
cubic  centimeter  of  alcohol  has  the  same  mass  as  one  of  water — indeed  it 
is  manifestly  untrue  if  quantity  of  matter  has  any  relation  to  either  inertia 
or  to  weight,  or  any  other  than  a  geometrical  significance. 

The  following  experiments  indicate  a  .logical  method  of  estimating 
mass : 

1.  Suppose  that  two  cubes  of  the  same  material,  of  exactly  the  same 
size,  are  successively  set  in  motion  on  a  smooth  horizontal  table  by  a  pull 
exerted  through  a  spring  balance.     It  will  be  found  that  the  spring  will 
be  stretched  by  the  same  amount  in  each  case  if  equal  accelerations  are 
imparted  to  the  two  masses.     With  a  given  cube  it  is  also  found  that  the 
elongation  of  the  spring  is  proportional  to  the  acceleration,  and  therefore, 
by  the  definition  of  force,  also  proportional  to  the  force.     If  another  cube 
of  twice  the  volume  be  used,  the  spring  will  be  stretched  twice  as  much 
in  imparting  the  same  acceleration.     In  this  case  one  cube  has  twice  the 
mass  of  the  other,   in  whatever  way  we  may  estimate  mass.     Were  all 
matter  alike  we  might  measure  its  mass  in  cubic  inches. 

2.  If  we  take  cubes  of  the  same  size,  but  of  different  substances,  the 
force    required   to    produce  the  same  acceleration  will    be  found   to    be 
different  in  each  case,  showing  that  inertia  is  not  proportional  to  volume 
if  we  compare  different  substances.     From  (1)  we  may  assume  that  if  the 
cubes  in  (2)  are  of  such  relative  sizes   that  the  same  applied  'force,   as 
measured  by  the  stretching  of  the  spring  will  produce  the  same  accelera- 
tion, that  is,  if  the  inertias  are  equal,  the  masses  are  equal. 

This  method  of  measuring  mass,  which  defines  it  in  terms  of  inertia, 
is  called  the  dynamical  method.  It  is  difficult  to  apply  practically.  The 
statical  method  of  weighing,  in  which  we  compare  masses  in  terms  of  the 
balanced  attraction  between  them  and  the  earth,  is  more  convenient. 
This  method  will  be  discussed  in  detail  later. 

Density  is  the  mass  of  unit  volume,  while  specific  gravity  is  the  ratio 
between  the  density  of  a  given  substance  and  that  of  water  taken  as  a 
standard.  The  numerical  value  of  the  density  of  a  substance  depends 
upon  the  system  of  units  adopted;  for  example,  in  the  English  foot- 
pound system  the  density  of  water  would  be  62.321.  Specific  gravity  is 
a  numerical  ratio,  and  independent  of  the  system  of  units. 


V 

DYNAMICS. 

FORCE   AND    MOTION    (DYNAMICS). 

17.  The  most  familiar  of  all  forces  is  that  of  gravitation,  and  it  is 
not  strange  that  it  was  the  first  to  be  investigated.  It  is  surprising,  how- 
ever, to  consider  how  many  centuries  elapsed  after  this  subject  became 
an  object  of  attention  before  any  one  was  able  to  accurately  describe  the 
way  in  which  a  body  falls.  In  their  philosophic  zeal  to  find  why  bodies 
fall,  the  ancient  Greeks  overlooked  the  simpler  problem  of  how  they  fall. 
They  explained  the  fall  of  bodies  by  saying  that  every  body  seeks  its 
proper  place,  and  that  the  place  of  heavy  bodies  is  below  that  of  light 
bodies.  It  was  to  them  a  logical  inference  that  a  very  heavy  body  would 
seek  its  place  faster  than  one  less  heavy.  It  seems  to  have  never  occurred 
to  them,  nor  to  any  one  until  many  centuries  later,  to  try  the  simple 
experiment  of  watching  bodies  fall,  and  thus  test  this  assumption.  There 
can  be  no  better  illustration  of  the  danger  of  a  priori  reasoning. 

Galileo,  a  professor  in  the  University  of  Pisa,  in  the  early  part  of  the 
seventeenth  century,  seems  to  have  been  the  first  to  abandon  the  vague 
methods  of  the  older  thinkers.  He  undertook  to  formulate  an  accurate 
description  of  how  bodies  fall,  and  in  so  doing  laid  the  foundations  of 
Dynamics,  or  that  branch  of  Mechanics  which  treats  of  matter  in  motion. 
He  found  by  dropping  bodies  from  the  Leaning  Tower  at  Pisa  that  the 
speed  of  fall  is  proportional  to  the  time  of  fall,  regardless  of  the  nature  or 
size  of  the  object,  provided  that  the  weight  is  sufficient  to  make  the 
resistance  of  the  air  negligible.  He  thus  introduced  the  idea  of  accelera- 
tion, or  change  of  velocity  per  unit  time.  He  also  perceived  a  very 
fundamental  principle  of  mechanics — that  a  body  may  be  subjected  to 
two  or  more  simultaneous  motions,  which  go  on  absolutely  independently. 
Before  his  time  it  seems  to  have  been  tacitly  held  that  one  effect  must 
cease  before  another  begins — that  a  projectile  must  cease  to  move  hori- 
zontally before  it  can  drop  vertically.  Galileo  was  the  first  to  show  that 
the  rate  of  vertical  fall  is  absolutely  independent  of  any  horizontal  motion 
which  the  body  may  possess,  as  any  one  may  see  in  the  case  of  a  stone 
projected  horizontally. 

These  ideas  were  further  elaborated  and  more  definitely  expressed  by 
Sir  Isaac  Newton  in  his  Principia  (London,  1687).  He  recognized  the 
property  of  mass  as  inherent  in  bodies,  and  considered  the  action  upon 
matter  not  only  of  gravitation,  but  of  all  forces  which  may  cause  motion. 
He  also  distinctly  formulated  the  principle  of  the  parallelogram  of  forces, 
which  had  been  more  or  less  vaguely  perceived  in  special  cases  by  others 
before  him. 

The  conclusions  of  Newton  may  be  summarized  substantially  as 
follows : 

1.  Every   body  tends  to   move  uniformly   in  a  straight  line  unless 
acted  upon  by  an  impressed  unbalanced  force. 

2.  Change    in    motion    (acceleration)    is    in    the    direction   of    the 
impressed  force  and  proportional  to  its  intensity.      This  statement  con- 
tains within  it  the  definition  of  force. 

3.  Action  and  reaction  are  equal  and  oppositely  directed. 

These  laws  are  not  the  results  of  a  priori  reasoning,  but  are  simply  a 
generalized  description  based  on  our  daily  experience.  Their  application 
to  cases  outside  of  our  experience  is  a  typical  example  of  scientific  induc- 
tion. As  a  matter  of  fact,  we  have  no  knowledge  of  any  body  which  has 
ever  moved  uniformly  in  a  straight  line;  but  we  see  that  all  bodies  -tend 


10  PROPERTIES    OF    MATTER. 

to  do  so  more  and  more  as  we  remove  opposing  forces,  and  the  conclu- 
sion of  the  first  law  of  Newton  seems  to  be  a  logical  necessity. (1) 

We  hear  much  of  the  impossibility  of  perpetual  motion.  As  a  matter 
of  fact,  perpetual  motion  is  the  rule  of  nature.  It  is  impossible  only  in 
the  limited  sense  that  a  body  cannot  overcome  resisting  forces,  thus  doing 
work,  without  being  supplied  with  energy  from  some  external  source,  or 
in  other  words  having  its  motion  replenished  at  the  expense  of  other 
moving  bodies. 

18.  Experiments  such  as  those  described  in  section  16  show  that  the 
acceleration  imparted  to  a  given  mass  varies  directly  as  the  force,   and 
that  the  acceleration  due  to  a  given  force  varies  inversely  as  the  mass 
acted  upon.     This  proportionality  is  expressed  as  follows : 

(4)  F=  Kma, 

K  being  a  numerical  constant  depending  on»the  system  of  units  adopted. 
This  is  the  algebraic  expression  of  Newton's  second  law. 

The  C.  G.  S.  unit  of  force,  called  the  dyne,  is  defined  as  the  force 
required  to  produce  in  one  second  a  change  of  velocity  of  one  centimeter 
per  second  in  a  mass  of  one  gram.  Under  these  conditions,  K  =  1. 

19.  Composition    and    Resolution   of   Motions,   Velocities, 
Accelerations,  and  Forces. — If  a  body  be  moved  from   A  to  B, 
thence  to  C,  its  final  position  is  the  same  as  though  it  had  been  moved 
directly   from    A   to    C.       Geometrically,    in    dealing   with   directed    or 
* '  vector ' '  quantities,  we  may  write, 

(5)  AB  +  BC  =  AC. 

If  AB  is  proportional  to  and  in  the  direction  of  a  given  velocity  vt 
(motion  per  unit  time),  and  BC  in  the  same  way  represents  another 
velocity  z/2,  then  the  line  AC  will  represent  the  resultant  velocity  v.  In 
other  words  if,  by  the  simultaneous  application  of  two  forces,  the  two 
velocities  vl  and  z/2  proportional  to  AB  and  BC  are  simultaneously 
produced,  the  body  will  actually  move  along  the  line  AC  with  a  resultant 
velocity  v  proportional  to  the  length  of  that  Tine. 

The  same  reasoning  applies  to  any  number  of  velocities.  If  lines 
representing  them  be  drawn  end  to  end,  the  closing  side  of  the  polygon 
thus  formed  will  be  the  resultant  velocity. 

Since  acceleration  is  change  of  velocity  in  unit  time,  the  same  rules 
apply  to  the  composition  of  accelerations;  and  also  to  forces,  since 
accelerations  are  proportional  to  forces. 

Suppose  that  vt  is  the  velocity  of  a  body  at  the  time  tfr,  v2  the  velocity 
(differently  directed)  at  the  time  4.  Draw  from  a  point  two  lines  AB, 
AC,  proportional  to  the  two  velocities  and  similarly  directed.  Then  BC 
represents  the  change  of  velocity,  both  in  amount  and  direction,  or 

(6)  ^-z^aa-O-BC 

This  is  always  geometrically  true  of  the  first  equation,  but  is  arith- 
metically true  only  when  the  velocities  are  in  the  same  direction;  it  is 
always  true  of  the  second  equation. 

20.  Resolution  of  Vector  Quantities.— By  a  converse  method, 
any  one  motion,  velocity,  acceleration,  or  force  may  be  resolved  into  any 
number  of  components  in  any  desired  direction. 

(1)  For  discussion  of  work  of  Galileo  and  Newton  see  Mach,  Science  of  Mechanics,  Ch.  II;  also 
articles  Galileo  and  Newton,  Ency.  Brit. 


r 


FORCE  —  MOMENTUM.  11 

In  general,  we  wish  to  resolve  these  quantities  into  two  components  at 
right  angles  to  each  other,  along  the  X  and  the  Y  coordinate  axes.  If  a 
force  F  have  a  direction  at  an  angle  a  with  the  X  axis,  and  A'  and  Y  be  its 
components, 

X  =  F  COS  a 

Y= 


All  these  results  are  necessary  consequences  of  the  definitions  of 
velocity,  acceleration,  and  force,  but  they  may  be  established  as  well  by 
direct  experiment. 

21.  The  moment  of  a  force   is   the   tendency   of  the  force   to 
produce  rotation  about  an  axis,  and  is  measured  by  the  product  of  the 
force  into  the  perpendicular  distance  from  the  axis  to  the  line  of  action  of 
the  force. 

22.  There  are  two  necessary  conditions  of  equilibrium  for  any  rigid 
body  : 

1.  The  algebraic  sum  of  the  forces  acting  at  any  point  of  the  body 
must  be  zero  for  both  the  X  and  the  Y  direction,  or  in  any  direction, 
otherwise  there  will  be  motion  in  that  direction, 

,7,  sAr=S^cosa  =  0 

2  Y=z  Fs'm  a  =  0. 

2.  The   algebraic  sum  of  the    moments   about  any    point  whatever 
must  be  zero,  or  there  will  be  rotation  about  that  point. 

2^  =  0 

S  Yx  =  0. 

Rotations  to  the  left  (anticlockwise)  are  taken  as  positive,  those  to  the 
right  as  negative,  because  angular  measurements  are  usually  regarded  as 
increasing  in  the  anticlockwise  direction. 

23.  Parallel    Forces.  —  If  R  be  the   resultant   of  a   number  of 
parallel  forces  acting  on  a  body, 

(9)  R  =  Fl  +  F2 


the  proper  algebraic  sign  being  given  to  each  force. 

The    moment  of  the   resultant  is   the  same   as    the  resultant  of  the 
moments,  or 

(10)  Rx  =  F,  x,  +  F2  xz  -f  .   .   .  etc., 

x,  x^  x2,  etc.,  being  measured  in  the  same  direction  from  any  common 
origin. 

24.     Momentum. — By  the  definition  of  force, 

F=  ma  =  1 
or 


(11) 

Experiment  shows  that  the  effect  of  a  blow  given  by  a  moving  mass  is 
proportional  to  the  mass  and  to  its  velocity.  The  product  mv  is  called 
momentum.  The  above  equation  shows  that  change  in  momentum  is 
proportional  to  the  force  acting  on  the  body  and  to  the  time  the  force  acts. 


12  PROPERTIES    OF    MATTER. 

QUESTIONS   AND    PROBLEMS. 

1.  Formulate  from  your  own  experience  some  physical  law  not  explicitly 
stated  in  the  text-books. 

2.  Rotate  a  cup  containing  water  or  coffee,  and  note  whether  the  liquid  moves. 
Explain  the  result. 

3.  A  wagon  moves  with  the  uniform  speed  s.     In  what  path  does  a  point  on 
the  circumference  move  relative  to  the  axle  ? — relative  to  the  earth  ?     What  is  the 
speed  of  the  top  point  of  the  wheel  relative  to  the  earth?—    f  the  bottom  point? 

4.  If  w  be  the  angular  velocity  of  a  point  with  radius  vector  r,  prove  that  the 
linear  speed  ^  =  rw. 

5.  The  wheels  of  the  wagon  make  10  revolutions  per  minute.     What  is  their 
angular  speed  per  second  with  reference  to  the  axle  ?     If  the  radius  is  50  cm.  what 
is  the  linear  speed  of  the  top  point  with  reference  to  the  lowest  point  ? — its  angular 
speed  ? 

6.  What  are  the  angular  speeds  of  the  minute  and  the  hour  hands  of  a  clock? 

7.  The  wagon  referred  to  above  starts  from  rest  with  a  uniform  acceleration, 
and  at  the  end  of  30  seconds  is  traveling  at  a  spee*d  of  4  meters  per  second.    What 
is  its  linear  acceleration  ? — the  angular  acceleration  of  the  top  point  with  reference 
to  the  axle  ? — with  reference  to  the  lowrest  point  ? 

8.  What  is  the  resultant  velocity  (speed  and  direction)  of  a  boat  rowed  at  a 
rate  of  4  miles  per  hour  eastward  and  drifting  at  a  rate  of  3  miles  per  hour 
southward  ? 

9.  A  mass  of  100  grams  is  at  one  instant  moving  eastward  with  a  speed  of 
50  cm.  per  second.     Five  seconds  later  it  is  moving  in  a  direction  60°  north  of  east 
with  the  same  speed.     What  is  the  direction  and  amount  of  its  acceleration? 

10.  In  the  above  case  what  force  in  dynes  acts  on  the  body? 

11.  A  force  of  50  pounds  weight  acts  vertically  downward  on  a  bicycle  when 
its  crank-arm  is  at  an  angle  of  45°  with  the  horizon.     What  is  the  component  force 
which  tends  to  cause  rotation  ?    What  does  the  other  component  do  ? 

12.  In  the  above  case,  if  the  crank-arm  is  six  inches  long,  what  is  the  moment 
of  the  force  about  the  axle  expressed  in  the  foot-pound  system  ? 

13.  A  force  of  1000  dynes  acts  for  10  seconds  on  a  mass  of  20  grams.     What 
is  its  change  in  velocity  ? 

14.  A  horizontal  meter  rod  is  acted  upon  in  an  upward  direction  by  the  forces 
200,  500,  300,  and  700  dynes,  at  distances  of  20,  25,  35,  and  60  cm.  from  one  end 
respectively,  and  by  a  downward  force  of  800  dynes  at  a  distance  of  40  cm.  from 
that  end.     What  is  the  resultant  force,  its  point  of  application,  and  the  resultant 
moment? 

15.  Why  is  a  heavy  fly-wheel  often  used  in  running  machinery? 

16.  Why  is  a  balance  weight  used  on  the  driving-wheel  of  a  locomotive 
opposite  the  crank-pin  ? 

GRAVITATION. 

25.  Acceleration.     Experience  shows  that  every  unsupported  body 
falls  to  the  earth  with  a  uniform  acceleration.      In  all  cases  where  bodies 
are  seen  to  rise — for  instance,  balloons — or  where  the  acceleration  is  not 
constant  and  the  same  for  all  substances,  the  effect  can  be  traced  to  the 
buoyancy  of  the  surrounding  medium,  or  to  some  resisting  force.     The 
force  acting  between  any  body  and  the  earth  is  called  the  weight  of  the 
body,  and  like  any  other  force  it  is  defined  by  the  relation 

(12)  W=mg,     ,.. 

yft+iu& 

where g  is  the  acceleration  due  to  rnnnlMjjfiii  (about  980  cm.  or  32  feet 

per  second  gain  of  velocity  per*s>econd). 

*  jtf^LifjLjFr 

26.  Center  of  QimvWp. — Each  element  of  a  given  mass  is  acted 
upon  independently   by  gravity.      (See  section  23.)     The  total  weight 
(resultant  force)  is  the  sum  of  the  elementary  weights,  or 

(13)  W=W^+W9  +    .     .     .     .     Z£V 


THE    BALANCE.  13 

The  moment  of  the  entire  weight  about  any  given  point  must  be  the 
same  as  the  sum  of  the  elementary  moments.  If  the  elements  are  at  the 
horizontal  distances  x^  xz  .  .  etc.,  from  the  point  considered, 

(14)  Wx  —  w^Xi  +  WtX^  .  .  .  etc. 

The  point  of  application  of  the  resultant  weight  must  lie  in  a  vertical 
line  at  a  horizontal  distance  from  the  origin  chosen,  where 

V  *  ffftt         \      /*/»•         I 


This  point  is  called  the  center  of  gravity,  or  of  weight.  The  body 
acts  with  reference  to  external  forces  as  though  its  entire  mass  were  con- 
centrated at  that  point.  If  the  body  be  suspended  by  a  string,  the  center 
of  gravity  will  lie  in  the  prolongation  of  the  string,  and  as  low  as  its  con- 
straints will  permit.  If  balanced  on  a  knife  edge,  the  center  of  gravity 
will  be  vertically  above,  at  or  below  the  knife  edge.  When  the  center  of 
gravity  is  above  the  point  of  support,  equilibrium  is  unstable;  when  coinci- 
dent with  it,  neutral;  when  below  it,  stable.  The  conditions  are  respec- 
tively similar  to  those  of  a  body  on  a  hill  top,  one  on  a  plain,  and  one  in 
a  valley. 

27.  The  Balance.  —  From  the  relation  W=  mg  we  may  determine 
the  mass  of  bodies  by  a  statical  method,  by  balancing  their  weights  from 
the  ends  of  an  equal-armed  lever.  In  general  it  is  impossible  to  secure 
lever  arms  of  exactly  equal  lengths.  There  are  several  conditions  upon 
which  the  sensitiveness  and  accuracy  of  a  balance  depend,  some  of  them 
inconsistent  with  each  other. 

Let  the  weight  of  the  beam  be  w,  and  its  center  of  gravity  at  a  dis- 
tance h  below  the  point  of  support.  P^  and  P2  are  the  weights  of  the 
scale  pans  and  their  appendages.  The  system  should  balance  with  no 
weights  on  the  pans.  If  the  arms  of  the  balance  are  of  lengths  at  and  a2 
the  condition  of  equilibrium  is 

(16)  Pt  a,  =  P2  a2 

If  the  weights  Wt  and  W2  be  added,  the  condition  of  equilibrium 
becomes 

(17)  (W,  +  Pt}a^(W,+P,}a, 

By  subtraction  of  (16)  from  (17) 

(18)  W1a1=lV2a2 

If  at  =  a2,  then  Wl  =  W2.  If  this  is  not  the  case,  and  if  Wt  is  the 
weight  of  the  standard  required  to  balance  the  unknown  weight  W  in 
one  pan,  and  W2  the  required  weight  when  the  object  is  placed  in  the 
other  pan,  we  have 

W,  at  =  W2  a2 

(19)  IV  a,=  W2  a2 

w=- 


This  is  Gauss's  method  of  double  weighing. 

The  conditions  for  a  good  balance  are  that  it  shall  be  true;  that  is  to 
say,  its  arms  horizontal  and  of  equal  length;  it  shall  be  stable;  it  shall  be 


14  PROPERTIES    OF    MATTER. 

sensitive;  it  shall  have  a  small  period  of  oscillation,  so  that  readings  may 
be  quickly  taken. 

The  sensitiveness  of  a  balance  is  proportional  to  the  deflection  of  the 
beam  produced  by  a  small  overweight  placed  in  one  pan;  usually  this 
overweight  is  one  milligram. 

Let  the  angle  between  the  axes  of  the  arms  (deflection  downward)  be 
2a.  If  the  overweight  x  deflects  the  beam  or  a  pointer  attached  to  it 
through  the  angle  8,  the  condition  of  equilibrium  is 

(20)      (  W+  x~}  a  .  cos  (a+0)  =  Wa  .  cos  (a— 0)  +  wh  .  sin  e 
Expanding  and  collecting  coefficients, 

[  (2  W-\-  x)  a  .  sin  a  +  w/i]  sin  e  =  ax  .  cos  a  .  cos  6 

ax  .  cos  a 

tan0  =  -— —   ,0  --_  ,          — -r— 
wk  +  (2  W+vc)  a  sin  a 

In  above  expressions,  2  W  includes  the  weight  of  the  pans. 
If  the  beam  be  straight  (by  which  is  meant  that  the  fulcrum  and  the* 
two  knife  edges  supporting  the  pans  are  in  the  same  straight  line), 

(22)  tan*  =4=^ 

/      wh 

From  this  expression  we  see  that  the  sensitiveness  is  greater  the 
greater  the  length  of  the  arms,  the  lighter  the  beam,  and  the  nearer  the 
center  of  gravity  of  the  beam  is  to  the  fulcrum.  It  is  entirely  independent 
of  the  load' W if  the  beam  is  straight.  To  preserve  this  condition  under 
all  loads  the  beam  must  be  rigid. 

Stability  and  quickness  of  vibration  are  directly  proportional  to  h\ 
consequently  it  is  impossible  to  secure  these  advantages  simultaneously 
with  great  sensitiveness.  A  compromise  is  necessary. 

Balances  are  made  for  which  a  sensitiveness  sufficient  to  show  differ- 
ences of  weight  of  one-thousandth  milligram  is  claimed  ;  probably  the 
limit  of  accuracy  is  about  .01  mg. 

For  accurate  determinations  of  mass  it  is  necessary  to  correct  for  the 
buoyancy  of  the  atmosphere.  A  delicate  balance  is  very  sensitive  to 
temperature  changes  and  to  air  currents,  and  must  be  protected  from 
them  when  readings  are  made. 

PROBLEMS. 

17.  A  meter  rod  balances  on  a  knife-edge  at  division  40  if  a  100  gram  weight 
hangs  from  it  at  division  32.     What  is  the  weight  of  the  rod  ? 

18.  From  a  uniform  circular  disc  of  20  cm.  radius  a  smaller  disc  of  10  cm. 
radius  is  cut  tangentially.    Where  is  the  center  of  gravity  of  the  remaining  portion  ? 

28.  Kepler's  Laws. — The  German  astronomer  Kepler,  about 
1609,  after  many  vain  attempts  to  formulate  simple  laws  descriptive  of  the 
motions  of  the  planets,  on  the  basis  of  Tycho  Brahe's  astronomical 
observations,  succeeded  in  establishing  three  such  laws,  which  have 
become  the  basis  of  mathematical  astronomy.  .They  are  as  follows  : 

1.  All  the  planets  move  in  elliptic  orbits,  with  the  sun  at  one  focus. 

2.  The  motion  of  each  planet  is  such  that  the  radius  vector  drawn 
from  it  to  the  sun  sweeps  over  equal  areas  in  equal  times. 

3.  The  squares  of  the  periods  of  rotation  of  the  planets  about  the 
sun  are  proportional  to  the  cubes  of  their  mean  distances  from  it. 


K 

UNIVERSAL    GRAVITATION — CENTRIPETAL    ACCELERATION.  15 

29.  Universal  Gravitation. — The  ancients  observed  that  bodies 
fall,  and  speculated  as  to  why  they  fall.  Galileo  was  the  first  to  observe 
how  they  fall.  About  the  middle  of  the  seventeenth  century,  several 
persons  suggested  the  possibility  that  the  moon  and  the  planets  were  held 
in  their  orbits  by  a  force  similar  to  that  of  weight,  but  Newton,  about 
1666,  was  the  first  to  clearly  recognize  this  law  of  attraction  as  possibly 
a  general  principle,  and  to  state  it  as  a  function  of  the  mass  and  the 
distance  apart  of  the  two  bodies.  The  form  which  he  gave  to  the  law 
was  this — 

(23)  f=J&2*. 


F  being  the  attractive  force,  mt  and  m2  the  masses  of  the  two  bodies, 
d  the  distance  between  them,  and  K  a  numerical  constant  depending  on 
the  system  of  units  adopted.  If  raT  be  the  mass  of  the  earth,  m2  that  of 
another  body  at  the  earth's  surface,  at  a  distance  r  from  the  earth's  center 
of  gravity,  and  JFthe  weight  of  the  second  body, 


(24) 


This  equation  was  simply  the  statement  of  a  hypothesis  until  it  was 
verified  indirectly  by  astronomical  observations  and  directly  by  laboratory 
experiments. 

30.  Centripetal  Acceleration.  —  The  orbits  of  the  moon  and  the 
planets  are  ellipses  differing  very  slightly  from  circles.  To  make  a  body 
follow  a  circular  path  by  constantly  drawing  it  from  the  tangential  path 
which  it  tends  to  follow  on  account  of  its  inertia,  the  constant  action  of 
some  force  is  required;  if  the  body  moves  at  a  uniform  rate  around  its 
orbit,  the  deflecting  force  must  always  act  at  right  angles  to  the  path,  so 
that  there  is  no  component  of  acceleration  in  that  path.  To  fulfill  this 
condition  for  a  circular  or  nearly  circular  path,  the  force  must  be  always 
directed  toward  the  center  of  the  circle,  and  the  acceleration  produced  by 
it  is  the  centripetal  acceleration.  To  calculate  this  acceleration,  consider 
an  element  of  the  orbit  so  short  that  it  may  be  considered  a  straight  line. 
In  a  very  short  time  /  suppose  that  the  body  deviates  from  a  tangent 
drawn  to  the  circle  at  its  original  position  by  a  distance  d,  owing  to  the 
central  attraction.  If  a  is  the  acceleration  produced  by  this  force,  the 
average  speed  in  a  radial  direction  during  the  time  /  is 


The  arc  traversed  is 

b  =  vt, 
Also,  from  similar  triangles, 

d  :  b  =  b   :  2r. 
Therefore, 

(26)  »=£. 

This  may  be  tested  experimentally  as  follows  :     Two  spheres  of  masses 
m^  and  m2  slide  on  a  smooth  rod  attached  to  a  whirling  table  at  right 


16  PROPERTIES    OF    MATTER. 

angles  to  its  axis  of  rotation.  If  these  two  masses  be  connected  by  a 
string  and  the  table  rotated,  it  will  be  found  in  general  that  the  connected 
masses  fly  off  to  one  end  or  the  other  of  the  rod;  but  if  placed  in  such  a 
position  that  the  centers  of  the  spheres  are  at  distances  r^  and  rz  from  the 
axis  of  rotation,  such  that  rt  :  r2  =  m2  :  mlt  it  will  be  found  that  the 
bodies  will  be  in  equilibrium  whatever  the  speed  of  rotation.  If  (26)  is 
true,  -  ^ 


or  if  ^  be  the  angular  velocity, 

v,  =  r, 

Substituting  above, 

(27)  m 

in  accordance  with  the  results  of  the  experiment. 

31.  Newton's  Proof.  —  If  the  moon  moves  in  a  nearly  circular  orbit 
about  the  earth  as  a  center,  with  a  nearly  uniform  velocity,  a  central 
attracting  force  is  necessary.  Newton  assumed  that  this  force  is  due  to 
the  attraction  of  the  earth,  and  that  it  varies,  just  as  the  intensity  of  light 
for  example,  inversely  as  the  square  of  the  distance.  If  R  be  the  radius 
of  the  moon's  orbit,  r  the  radius  of  the  earth,  g  the  acceleration  of 
gravity  at  the  earth's  surface,  and  gt  the  acceleration  of  gravity  at  the 
distance  of  the  moon, 


By  the  law  of  centripetal  acceleration  we  also  have 


Newton  tested  these  relations  by  substituting  values  of  v  and  R  from 
the  imperfect  data  then  obtainable;  the  equality 

S 

did  not  appear  to  be  verified.  Many  years  later  the  distance  of  the  moon 
was  recomputed  by  Picard;  Newton  substituted  the  new  values  in  his 
equation,  and  found  it  completely  verified. 

Newton  likewise  found  that  Kepler's  laws  were  in  almost  exact  accord 
with  the  hypothesis  of  a  central  attractive  force,  varying  inversely  as  the 
square  of  the  distance.  Any  apparent  deviations  from  Kepler's  laws  are 
explained  by  the  perturbing  influence  of  other  attracting  bodies.  The 
position  of  the  planet  Neptune  was  almost  exactly  computed  from  its 
perturbations  on  Uranus,  by  Adams  and  by  Leverrier,  before  it  had  ever 
been  seen,  although  these  perturbations  had  the  effect  of  displacing 
Uranus  by  only  2'  of  arc.  Galle,  of  Berlin,  pointed  his  telescope  in  the 
direction  assigned  by  Leverrier,  and  within  half  an  hour  discovered  the 
planet.  The  accurate  fulfillment  of  astronomical  predictions  is  a  constant 
verification  of  Newton's  law  of  universal  gravitation. 


.   DIRECT    VERIFICATION. 

32.  Direct  Verification. — The  resultant  attraction  of  the  ear 
would  be  exactly  directed  toward  its  center  if  it  were  of  uniform  density, 
or  if  its  density  varied  uniformly  between  its  center  and  circumference. 
As  a  matter  of  fact,  local  irregularities  may  cause  measurable  deflections 
of  a  plumb  line.  In  1774  Maskelyne,  the  astronomer  royal  of  England, 
determined  the  deviation  of  a  plumb  line  produced  by  Mount  Schehallien, 
in  Scotland.  The  mountain  is  of  very  regular  shape,  and  a  rough  calcu- 
lation of  its  density  could  be  made  by  a  study  of  its  geological  structure. 
From  these  data  its  mass  and  the  position  of  its  center  of  gravity  could 
be  roughly  determined  It  was  found  by  comparison  with  the  directions 
of  the  stars  that  the  deflection  of  the  plumb  bob  toward  the  mountain  was 
about  6"  from  two  opposite  stations  on  the  north  and  the  south.  If  the 
acceleration  toward  the  mountain  be  g^ 

(29)  £1-=*   =tana 

o 

Also,  if  mt  be  the  mass  of  the  earth,  m2  that  of  the  mountain, 


(30) 


mn      dz  tan  a 


We  can  thus  calculate  the  earth's  mass  as  compared  with  that  of  the 
mountain.  Thfc  s  number  is  inconveniently  large,  so  the  results  are  usually 
expressed  in  terms  of  the  earth's  density,  from  the  relation 


(31)  mt 

This  method,  which  is  evidently  liable  to  great  inaccuracies,  gave 

(32)  P  =  4.71 

Henry  Cavendish,  in  1798,  secured  more  trustworthy  results  by  the 
use  of  the  torsion  balance.  Two  lead  spheres,  st  and  s2,  about  two  inches 
in  diameter,  were  attached  to  the  ends  of  a  light  wooden  rod  several  feet 
long.  This  was  suspended  horizontally  from  a  fine  wire,  and  the  entire 
system  enclosed  in  a  glass  case.  Two  large  spheres,  St  and  S2,  were 
attached  to  a  frame  so  that  they  could  be  brought  to  any  required  position 
with  respect  to  st  and  s2.  In  one  position  the  torsion  arm  would  be 
twisted  in  one  direction;  in  another  position,  in  the  opposite  direction. 
The  deflections  were  read  with  a  telescope.  The  magnitude  of  the  deflect- 
ing force  could  be  expressed  in  terms  of  the  known  force  required  to  twist 
the  wire  through  1°,  and  the  actual  angular  deflection  produced  by  the 
attraction  of  the  large  spheres.  The  density  of  the  earth  as  found  by 
Cavendish  was 

(33)  P  = 


18  PROPERTIES    OF    MATTER. 

Professor  Vernon  Boys,  of  England,  in  1894  devised  a  much  more 
sensitive  and  accurate  torsion  balance.  The  small  spheres  were  of  gold, 
about  5  mm.  in  diameter,  and  were  hung  by  fine  quartz  fibers  from  a  short 
torsion  beam  2.3  cm.  long,  which  was  itself  suspended  by  a  quartz  fiber, 
within  a  closed  cylinder.  Within  a  concentric  cylinder  which  could  be 
rotated  about  the  first  were  hung  two  lead  spheres  about  10  cm.  in  diam- 
eter. The  pairs  of  attracting  spheres  were  in  different  horizontal  planes, 
in  order  to  eliminate  as  far  as  possible  the  effect  of  each  large  sphere  on 
the  more  distant  small  sphere.  The  quartz  fibers  were  used  because  of 
their  small  torsional  rigidity  and  their  constancy  of  zero  position.  Boys 
was  able  to  show  the  attraction  between  two  small  shot,  the  actual  magni- 
tude of  this  force  being  about  1/200,000  dyne.  He  found  that 


P  =  5.527 

Richarz  and  Krigar-Menzel  (1898)  used  a  balance  method.  A  large 
cube  of  lead,  weighing  about  100,000  kilograms,  was  placed  under  a  sensi- 
tive balance  which  had  four  pans,  two  above  the  block  and  two  below  it, 
suspended  by  fine  wire  passing  through  holes  in  the  block.  The  process 
of  double  weighing  was  repeatedly  applied  to  two  approximately  equal 
masses,  m^  and  m2,  first  one  above,  the  other  below,  then  interchanged. 
The  apparent  weight  of  the  upper  mass  would  be  increased,  that  of  the 
lower  lessened  by  the  attraction  of  the  lead  block.  The  attraction  between 
the  block  and  a  kilogram  mass  was  about  1.4  milligram  weight,  which 
indicates  how  much  care  was  necessary  in  taking  the  readings.  They 
found 

A^=6.685X  10-8 

P-5.505 

t 

The  discrepancy  between  the  results  of  Boys  and  of  Krigar-Menzel 
and  Richarz  maybe  due  in  part  to  local  differences  of  density  of  the  earth. 

33.  Variations  of  the  Force  of  Gravitation.  —  Mass  is  an 
invariable  property  of  any  body;  its  weight  depends  on  the  mass  and  dis- 
tance of  another  body.  At  some  point  between  the  earth  and  the  moon, 
for  example,  an  object  might  have  no  resultant  weight  ;  but  its  mass 
would  remain  unchanged,  and  could  be  determined  by  a  dynamical 
method. 

The  weight  of  a  body  varies  at  different  points  of  the  earth's  surface, 
for  several  reasons: 

1.  There  are  local  and  irregular  variations  of  g  due  to  mountains, 
masses  of  rock  beneath  the  surface,  etc.,  which  cannot  be  computed. 

2.  The  earth  is  not  a  perfect  sphere,  but  is  flattened  at  the  poles,  so 
that  its  cross-section  is  approximately  an  ellipse,  the  minor  axis  coinciding 
with  the  axis  of  rotation.      If  a  body  be  carried  from  the  equator  to  the 
pole,  it  is  constantly  approaching  the  center  of  the  earth.     In  the  case  of 
a  spheroid  such  as  the  earth  there  is  no  definite  center  of  gravity,  toward 
which  the   plumb   line  would   point  wherever  it  might  be,  so  that  the 
increase  of  g  cannot  be  simply  calculated  in  terms  of  the  change  of  the 
earth's  radius.     If  the  latitude  is  X,  and  g&  the  value  of  g  at  the  equator, 
it  is  found  that  . 

(36)  *'  (!+- 


VARIATIONS    OF    THE    FORCE    OF    GRAVITATION THE   TIDES.          19 

3.     At  a  given  latitude  there  are  changes  of  g  with  the  altitude  above 
sea  level.     We  have 


g.     (r+*r 

or 

(37)  g^  =  (  1-y-)  ^-.  =  (1-0.000000314^)^-0 

This  applies  strictly  only  to  values  observed  on  a  high  tower  or  in  a 
balloon.  The  value  of  g  on  a  mountain  or  plateau  is  influenced  by  local 
masses. 

4.  In  order  to  hold  bodies  in  their  places  on  the  surface  of  the  earth 
a  centripetal  acceleration  is  necessary.  At  the  latitude  X  the  necessary 
acceleration,  in  a  plane  at  right  angles  to  the  polar  axis,  is  (assuming  the 
earth  to  be  spherical) 

(38)  —  =  rx<»*  =  ^rcos\ 

The  necessary  acceleration  is  furnished  by  gravity,  but  of  course  not 
in  the  line  of  action  of  gravity.  Let  G  represent  the  direction  and 
intensity  of  gravitation  if  the  earth  were  at  rest,  a  the  centripetal  acceler- 
ation with  the  earth  in  motion,  and  g  the  actual  observed  acceleration 
after  a  component  of  G  has  been  taken  to  furnish  the  necessary  centripetal 
acceleration.  From  the  above  equation  it  is  easy  to  calculate  the  relation 

(39)  g  =  G  (1  —  7^  cos  2X) 

Combining  above  equations  and  substituting  for  the  value  of  g  at  the 
equator  that  at  latitude  45°,  (980.61)  the  value  of  g  at  any  latitude  X  and 
height  h  above  sea  level  may  be  expressed  by  the  equation 

(40)  g  =  980.61  (1  -  0.00259  cos  2  x  -  0.0000003/z) 

It  is  the  residual  component  g  which  determines  the  weight  of  bodies. 

After  the  direct  pull  of  gravitation  has  lost  a  component  to  supply  the 
necessary  centripetal  acceleration  at  a  point,  the  remaining  component  is 
of  course  no  longer  directed  toward  the  earth's  center  except  at  the 
equator  and  the  poles;  only  in  these  places  does  a  plumb  line  point 
exactly  toward  the  center  of  the  earth. 

The  surface  of  a  liquid  is  always  normal  to  the  resultant  force  acting 
on  it,  and  the  ocean  is,  therefore,  when  not  disturbed  by  wind,  always 
normal  to  a  plumb  line.  Were  the  earth  to  cease  rotating,  the  water 
would  rush  from  the  equator  to  the  poles,  assuming  a  spherical  surface. 
Renewal  of  rotation  would  restore  the  present  shape.  We  can  thus  realize 
how  the  earth  was  flattened  at  the  poles  when  in  a  semi-molten  state. 
The  residual  component  of  gravitation  has  a  component  tangential  to  the 
earth's  surface  which  keeps  water  and  other  objects  from  rolling  down  the 
13-mile  hill  to  the  poles. 

34.  The  Tides  in  the  earth's  oceans  are  caused  by  the  attraction  of 
the  moon  and  the  sun.  The  component  due  to  the  former  is  more  than 
twice  as  great  as  that  due  to  the  latter,  on  account  of  the  nearness  of  the 
moon.  At  the  times  of  new  moon  and  full  moon  both  effects  are  in  the 
same  phase  and  the  tides  arehjgjiest  (spring  tide).  At  half-moon  the 

RA«^ 

OF  THE 

1IN1IVFRSITY 


20  PROPERTIES     OF    MATTER. 

crest  of  one  tidal  wave  coincides  with  the  trough  of  the  other,  and  the 
tides  are  low  (neap  tide). 

There  are  two  tidal  waves,  one  toward  the  side  of  the  moon  (but  not 
directly  under  it  on  account  of  a  lag  or  lead  due  to  the  solar  action,  fric- 
tion, etc.)  and  one  away  from  it.  To  explain  this  we  must  consider  the 
nearer  and  further  masses  of  water  and  the  intermediate  solid  earth  to 
behave  as  three  bodies.  The  nearer  is  more  accelerated  toward  the  moon 
than  the  earth,  the  further  is  less  accelerated  ;  and  so  they  move  apart, 
the  nearer  body  of  water  being  heaped  up,  and  the  earth  leaving  the 
further  body  behind.  Both  the  earth  and  the  moon  are  rotating  about 
their  common  center 'of  gravity  (situated  within  the  earth).  Looking  at 
the  problem  in  another  way,  we  can  consider  the  tides  as  arising  from  the 
centrifugal  tendencies  of  the  earth  and  water  with  respect  to  the  axis 
passing  through  the  center  of  gravity  of  the  two  bodies  regarded  as  one 
system.  If  the  moon  were  at  rest  with  respect  to  the  earth  the  results 
would  be  different  (see  Problem  26).  There  is  necessarily  a  loss  of 
energy  by  friction  between  tidal  waves  and  the  solid  earth,  and  when  the 
wave  is  stopped  by  impact  against  the  shore;  this  must  be  borrowed  from 
the  earth's  energy  of  rotation,  hence  the  days  must  gradually  be  growing 
longer. 

The  height  of  the  tides  is  very  different  in  different  places.  In  the 
open  Pacific  it  is  only  2  or  3  feet.  In  the  North  Sea  interference  of  two 
tidal  waves,  one  from  north  of  Scotland,  the  other  through  the  English 
Channel,  minimizes  the  effect  to  a  few  inches.  In  nearly  enclosed  seas 
like  the  Mediterranean  they  are  only  a  few  inches  high.  South  of  England, 
at  St.  Malo  and  the  Isle  of  Jersey,  they  are  some  35  feet  high,  and  in 
narrow  bays  such  as  the  Bay  of  Fundy,  into  which  large  masses  of  water 
are  driven  by  their  momentum,  the  tide  may  rise  to  70  feet  or  more. 

References. — Newton,  Kepler,  Tides — Mach,  Science  of  Mechanics;  Lodge, 
Pioneers  of  Science  ;  Ball,  Time  and  Tide.  Verification  of  law  of  gravitation — 
Mackenzie,  Laws  of  Gravitation ;  Tait,  Properties  of  Matter  ;  Burgess,  Physical 
Review,  May,  1902  ;  Boys,  Nature,  August  2,  9,  23,  1894,  and  Phil.  Trans.  Royal 
Soc.,  1895;  Poynting  and  Thomson,  Properties  of  Matter;  Poynting,  Scientific 
American  Supplement,  May  2,  1903. 

QUESTIONS   AND   PROBLEMS. 

19.  Derive  the  expression  for  centripetal  acceleration  by  applying  the  method 
of  section  19,  equation  (6). 

20.  Do  you  know  of  any  case  of  centrifugal  acceleration  or  force  ? 

21.  A  mass  of  lead  weighing  100  kilograms  has  its  center  of  mass  100  cm. 
above  the  pan  of  a  balance.     How  much  will  it  diminish  the  weight  of  100  grams 
in  the  pan  ? 

22.  If  the  earth  rotated  seventeen  times  as  fast  as  it  does,  what  would  be  the 
value  of  g  at  the  equator  ? 

23.  If  the  earth  rotated  more  than  seventeen  times  as  fast  as  it  does,  and  if 
we  were  unaware  of  its  rotation,  how  would  we  be  apt  to  state  the  law  of  gravita- 
tion ?    Would  it  be  correctly  stated  ? 

24.  Explain  the  trade  winds  and  cyclones.       Show  that  a  projectile  will  be 
deviated  to  the  right  in  the  northern  hemisphere  no  matter  in  what  direction  in  a 
horizontal  plane  it  may  be  fired. 

25.  If  a  projectile  could  be  weighed  while  moving  east  or  west  would  it 
weigh  the  same  as  if  at  rest  ? 

26.  If  the  moon  and  the  earth  had  no  mutual  acceleration,  how  would  the 
tides  be  affected  ? 

27.  What  shape  is  assumed  by  the  water  surface  in  a  cylindrical  vessel  when 
the  latter  is  rotated  rapidly?     Explain. 

28.  Explain  the  action  of  the  Watt's  governor  for  steam  engines. 

29.  Explain  the  "  centrifugal  "  cream  separator. 


DETERMINATION    OF   G  -  FALLING    BODIES.  21 

35.  Determination  of  g.  —  Consider  a  smooth  sphere  rolling 
down  a  smooth  inclined  plane  at  an  angle  of  a  with  the  horizon.  The 
resultant  acceleration  down  the  plane  is  the  resolved  part  of  g  in  that 
direction,  or 

(41)  a=g  sin  a. 

In  this  way  a  may  be  made  so  small  as  to  be  conveniently  measured, 
and  g  computed  from  the  above  relation.  This  method  was  used  by 
Galileo,  but  gives  only  crude  results. 

The  most  accurate  method  is  by  means  of  pendulum  observations. 
Consider  a  conical  pendulum  —  that  is,  a  simple  pendulum  which  rotates  in 
a  circle  of  radius  a,  so  that  the  supporting  cord  describes  a  cone  in  space. 
The  tension  in  the  cord  must  be  such  that  it  will  furnish  a  vertical  com- 
ponent to  support  the  weight  of  the  pendulum,  and  a  centripetal  force 
holding  the  pendulum  in  its  orbit.  We  have  evidently,  if  /  be  the  length 
of  the  pendulum,  and  h  the  height  it  is  raised, 

v2  j  a  :  g  =  a  :  I—  h. 

If  the  displacement  is  so  small  that  h  may  be  neglected  in  comparison 
with  /,  this  gives 


If  T  be  the  period  of  rotation  of  the  conical  pendulum, 


(42) 


Observation  shows  that  if  the  same  pendulum  vibrates  in  a  plane  the 
period  will  be  exactly  the  same  —  in  fact  we  shall  see  later,  in  discussing 
vibratory  motion,  that  the  circular  rotation  is  simply  the  resultant  of  two 
plane  vibrations  at  right  angles  to  each  other,  and  necessarily  takes  place 
in  the  same  time  as  either  vibration.  Furthermore,  for  all  small  displace- 
ments or  amplitudes  a,  such  that  h  may  be  neglected,  the  period  is 
independent  of  the  amplitude  a.  This  fact  was  discovered  by  Galileo. 

From  the  above, 

47T2/ 


If  the  pendulum  bob  is  small  and  heavy,  this  gives  a  very  accurate 
method  of  determining  g,  since  both  T  and  /  can  be  very  exactly 
measured.  In  the  case  of  the  compound  pendulum  (a  massive  bar),  the 
problem  is  not  so  simple,  since  each  element  of  the  pendulum  tends  to 
vibrate  at  a  different  rate.  There  are  relations,  however,  which  make  it 
possible  to  determine  the  length  of  the  equivalent  ideal  simple  pendulum, 
so  the  compound  or  Kater  pendulum  is  more  frequently  used. 

Some  values  of  g  are  as  follows  : 

Latitude..     0°         15°        30°        45°        60°       75°        90° 
g  ...........  978.1    978.4    979.4    980.6    981.9    982.7    983.1 

36.  Falling  Bodies.  —  Within  observable  distances  from  the  surface 
of  the  earth  the  acceleration  of  gravity  is  practically  constant,  or  the  gain 
in  velocity  per  second  is  the  same  during  each  successive  second.  If  v0  be 


22  PROPERTIES     OF     MATTER. 

the  initial  velocity  of  a  falling  body,  and  it  falls  from  time  t0  to  time  £x 
or  t  seconds, 

(44)  g= 


(45)  v,= 

Since  the  velocity  changes  at  a  uniform  rate,  the  average  velocity  is 

f  Ad\  ^t+^o 

(46)  v  =    —  —  . 


In  falling  from  the  height  h0    to  ht , 

_  Vt  +  2701 

(47)  *.-A-«f-     -^- 

Multiplying  (44)  by  (47),  we  get 

This  relation  is  perfectly  general ;  it  applies  to  bodies  projected 
upward  with  velocity  v0,  as  well  as  to  falling  bodies.  If,  however,  g  and 
distances  fallen  through  be  counted  as  positive,  the  velocity  of  upward 
projection  and  the  height  of  ascent  must  be  counted  as  negative. 

37.  Inclined    Fall. — Consider  a  body  rolling   down  an  inclined 
plane.     The  acceleration  a  is  found  by  the  relation, 

(49)  a  —  —f — £-  and  also  a  =  - 

v~~  *t  '  * 

The  average  velocity  down  the  plane  is 

(50)  v  =  ^+^ 

/"  f  "\  \  ^+       I        7^«   r*>    * 

/.-/,=  *=  -^—  * 

The  speed  acquired  by  falling  depends,  therefore,  solely  on  the  vertical 
distance  through  which  the  body  has  fallen  ;  but  the  direction  of  the 
velocity  is  determined  by  the  constraint  at  right  angles  to  the  direction  of 
motion. 

The  same  considerations  apply  to  the  pendulum ;  its  velocity  at  any 
point  is  that  due  to  the  fall  through  the  height  h0 —  hi.  In  passing 
through  its  resting  point,  however,  the  actual  motion  is  entirely  in  a 
horizontal  direction. 

38.  Nature  of  Gravitation. — No  reasonable  hypothesis  has  ever 
been  found  to  account  for  gravitation.     Le  Sage  explained  it  as  the  result 
of  the  bombardment  of  etherial  particles  against  bodies,   driving  them 
together;  but  this  would  imply  that  a  flat  body  would  be  heavier  when 
presented  broadside  than  when  presented  edgewise  toward  the  earth,  and 


f~  OF  THE  \ 

UNIVERSITY) 

CONSERVATION    OF    MASS.  28 

would  also  involve  a  continuous  expenditure  of  energy.  There  is  no  way 
of  shielding  matter  from  its  effects;  were  this  possible,  an  inexhaustible 
supply  of  work  might  be  obtained  by  so  shielding  half  of  a  vertical  wheel, 
which  would  be  driven  by  the  pressure  of  gravitation  on  the  unshielded 
half. 

39.  Conservation   of  Mass. — There  is  no  external  resemblance 
between  metallic  sodium,  the  violet  sodium  vapor  at  high  temperatures, 
and  the  glowing  yellow  sodium  vapor  at  still  higher  temperatures.     A 
chemical  compound,  such  as  sodium  chloride,  has  no  resemblance  what- 
ever to  either  of  its  constituents.     It  is  rather  doubtful  whether  we  can 
say  in  such  cases,  as  is  usually  done,  that  in  all  these  changes  the  matter 
remains  unchanged.     The  total  mass,  however,  does  seem  unalterable  by 
any  physical  change  or  chemical  combination.     Some   experiments  by 
Landolt  and  others  on  the  weight,  of  substances  in  a  sealed  tube  before 
and  after  combination  seem  to  indicate  the  possibility  of  a  very  slight  loss 
in  weight;  but  such  measurements  are  difficult  and  the  results  of  doubtful 
validity  when  compared  with  the  enormous  number  of  chemical  determi- 
nations showing  the  contrary. 

QUESTIONS   AND   PROBLEMS. 

30.  Draw  a  curve  with  the  values  of  latitude  and  g  given  in  section  35  as 
coordinates,  and  by  interpolation  find  the  value  of  g  in  Berkeley. 

31.  Calculate  the  length  of  the  seconds  pendulum  in  Berkeley. 

32.  How  would  a  large  block  of  lead  under  a  pendulum  affect  its  period  ? 

33.  If  a  seconds  pendulum  is  constrained  to  swing  in  a  plane  at  an  angle  of 
45°  with  the  vertical,  what  will  be  its  period  ? 

34.  A  body  is  thrown  upward  with  a  velocity  of  10  meters  per  second.     How 
far  will  it  rise  ? 

35.  How  long  will  it  be  before  the  body  in  preceding  problem  returns  to  the 
earth  ? 

36.  A  sphere  rolls  from  the  top  edge  to  the  bottom  of  a  hemispherical  bowl 
of  50  cm.  radius.     What  will  be  its  velocity  at  the  bottom? 

WORK   AND   ENERGY. 

40.  When  we  raise  a  weight  or  in  any  other  way  move  an  object 
against  external  forces,  we  say  that  we  do  work.     Not  only  is  the  idea  of 
force  involved,  but  the  idea  of  motion  as  well.     A  weight  lying  passively 
on  the  earth  does  no  work;  when  by  sinking  it  drives  the  wheels  of  a 
clock,  it  does  work.     A  hod- carrier  who  carries  a  load  of  brick  to  the 
third  story  of  a  house  evidently  does  twice  as  much  work  as  though  he 
carried  it  to  the  second  story.     Work  done  iipon  a  body  may  be  quanti- 
tatively defined  as  being  equal  to  the  product  of  force  overcome  by  the 
distance  through  which  the  displaced  body  is  moved.     The  work  done  by 
a  given  force  (the  product  of  the  force  and  the  distance  through  which  it 
acts)  is  in  general  greater  than  that  done  upon  any  one  resisting  force ; 
there  are  usually  unknown  frictional  forces  to  be  overcome  as  well,  or  the 
excess  of  applied  force  may  not  do  useful  work,  but  produce  acceleration 
of  the  body  moved.      A  moving  object  may  transfer  its  motion  in  part  or 
wholly  to  another  body  when  it  strikes  it ;  so  the  possession  of  motion 
implies  the  power  of  doing  work.     A  body  at  rest  may  also  under  certain 
conditions  have  the  power  of  doing  work.     A  raised  weight,  for  example, 
or  a  coiled  spring  may  set  a  clock  in  motion  when  the  escapement  is 
released.     The  possibility  of  work  in  such  cases  is  a  consequence  of  the 
position  or  the  state  of  a  working  body.     A  body  which  for  any  reason 


24  PROPERTIES     OF    MATTER. 

is  able  to  do  work  is  said  to  possess  energy;  this  energy  is  called  kinetic 
if  it  depends  upon  motion;  potential  if  it  depends  upon  position  or  state. 
The  total  energy  of  a  body  may  be  part  kinetic,  part  potential.  There 
are  other  classes  of  energy;  that  of  heat,  for  example,  which  seems  on 
analysis  to  be  due  to  molecular  kinetic  energy;  and  electrical  and  chemical 
energy. 

If  a  body  of  mass  m  is  raised  to  a  height  k0  ,  the  required  work  is 

W=  mgh0 

and  this  represents  the  potential  energy  of  the  body  at  this  height.     If 
the  body  falls  freely  to  the  height  hlf  we  have  from  equation  (48) 


Multiplying  by  m, 

(53)  4  m  (y*  —  v<?~)  =  mg  (h0  —  h^ 

The  right-hand  member  of  this  equation  represents  the  potential  energy 
of  the  mass  m  with  respect  to  the  height  /zx  when  raised  to  the  height  h0y 
or  the  work  expended  in  raising  it  ;  the  equivalent  left-hand  member, 
expressed  in  terms  of  velocity,  represents  the  kinetic  energy  gained  by 
freely  falling  from  hQ  to  h^. 

The  above  equation  may  also  be  written 

(54)  \  mv<?  +  mgh*  =  \  mv*  +  mgh^ 

which  shows  that  the  sum  of  the  potential  and  kinetic  energies  is  constant. 
This  is  ^true  for  oblique  as  well  as  for  vertical  fall,  as  illustrated  by  the 
case  of  'the  pendulum.  (See  equation  52.) 

41.  The  energy  of  a  vibrating  pendulum  is  all  potential  when  it  is 
at  the  end  of  its  swing,  all  kinetic  when  it  passes  through  its  resting 
point.  Its  kinetic  energy  in  the  latter  case  is  sufficient  (neglecting  the 
frictional  resistance  of  the  air)  to  carry  it  to  a  height  equal  to  that  through 
which  it  fell.  If  the  bob  is  elastic  and  if  it  strikes  another  similar  pendu- 
lum bob  of  the  same  mass,  the  first  will  come  to  rest,  the  second  will  move 
off  with  the  same  velocity  that  the  first  had  just  before  impact.  If  it  be  of 
different  mass  it  will  impart  some  of  its  kinetic  energy  to  it.  A  moving 
body  can  thus  pass  its  kinetic  energy  in  whole  or  part  to  another  body. 

For  a  level  surface  h  is  constant,  and  on  such  a  surface  no  work  can 
be  done  against  gravity.  Such  a  surface  is  called  an  equipotential  surface, 
for  wherever  a  body  may  be  placed  on  it  the  potential  energy  will  be  the 
same,  and  the  body  will  be  in  a  state  of  neutral  equilibrium  so  far  as 
gravity  is  concerned. 

A  raised  weight  tends  to  fall  ;  a  stretched  spring  will  fly  back  when 
released  ;  a  compressed  gas  tends  to  expand.  A  great  number  of  such 
instances  show  us  that  potential  energy,  whatever  its  form,  always  tends 
to  a  minimum.  Potential  energy  involves  potential  motion,  which  will  be 
realized  unless  checked  by  some  external  force.  A  body  is  in  stable 
equilibrium  when  its  potential  energy  is  a  minimum  under  its  conditions; 
consider  for  example  a  pendulum  at  rest.  It  is  in  unstable  equilibrium 
when  its  potential  energy  is  a  maximum  ;  for  example,  a  body  balanced 
on  a  hill-top.  It  is  in  neutral  equilibrium  when  its  possible  displacements 
cannot  change  its  potential  energy;  for  example,  a  sphere  on  an  equi- 
potential (level)  surface. 


ELEMENTS.  25 

When  a  falling  body  strikes  the  earth  and  is  brought  to  rest,  it  has 
lost  both  its  potential  and  its  kinetic  energy.  Examination  will  show, 
however,  that  its  temperature  has  been  raised.  The  rise  of  temperature 
of  a  given  body  will  be  found  to  be  proportional  to  the  distance  of  fall. 
The  quantity  of  heat  gained,  measured  in  terms  of  mass,  specific  heat, 
and  rise  of  temperature,  will  be  found  to  depend  only  upon  the  mass  of 
the  body,  without  regard  to  its  nature,  and  upon  the  distance  fallen. 
The  kinetic  energy  of  the  body  depends  upon  the  same  factors,  and  we 
might  infer  that  the  kinetic  energy  of  the  mass  has  simply  been  trans- 
formed to  kinetic  energy  of  its  smallest  particles  or  molecules.  This  is 
not  the  place  to  discuss  the  matter  in  detail,  but  later  we  shall  examine 
the  evidence  which  has  made  us  believe  that  heat  is  one  of  many  forms  of 
energy,  and  that  energy  may  be  changed  from  one  form  to  another  with- 
out any  loss  of  its  total  amount.  This  is  the  principle  of  the  conservation 
of  energy,  which  has  played  such  an  important  part  in  Physics  during  the 
past  half  century. 

The  unit  of  energy  is  the  erg,  or  work  done  by  a  force  of  one  dyne 
acting  through  one  centimeter.  Activity  is  rate  of  doing  work. 

References. — Stewart,  Conservation  of  Energy;  Mach,  Popular  Scientific  Lec- 
tures— On  the  Principle  of  the  Conservation  of  Energy;  Tait,  Recent  Advances  in 
Physical  Science. 

QUESTIONS  AND  PROBLEMS. 

37.  Attach  a  weight  to  a  spring  balance  and  drop  both.     Does  the  weight 
continue  to  elongate  the  balance  while  falling? 

38.  A  weight  is  placed  on  a  block  of  lead  and  both  are  dropped  freely. 
Would  you  expect  a  pressure  to  exist  between  them  ?    Would  the  case  be  altered 
if  the  weight  were  dropped  with  a  thin  horizontal  sheet  of  wood  under  it? 

39.  A  solid  and  a  hollow  sphere  of  metal  of  the  same  size  are  dropped  simul- 
taneously.    Will  their  accelerations  be  exactly  the  same? 

40.  Prove  that  the  energy  expended  is  never  less  than  the  potential  energy 
gained  (useful  work)  in  case  of  the  lever,  wheel  and  axle,  and  inclined  plane.     In 
cases  where  the  latter  is  less  than  the  former,  show  where  the  missing  energy 
has  gone. 

SPECIAL  PROPERTIES  OF  MATTER. 

42.  Elements. — Over  seventy  kinds  of  elementary  matter  are  known 
to  us,  and  from  time  to  time  others  are  discovered.  The  most  recent 
discoveries  are  the  atmospheric  gases  argon,  discovered  by  Lord  Rayleigh, 
neon,  xenon  and  krypton,  discovered  by  Ramsay,  and  the  gas  helium, 
also  discovered  by  Ramsay  in  the  mineral  cleveite.  It  is  interesting  to 
note  that  the  latter  element  was  known  hypothetically  many  years  pre- 
viously, from  a  bright  yellow  line  in  the  spectrum  of  the  sun's  chromo- 
sphere, which  could  not  be  identified  with  the  lines  of  any  known  element — 
hence  the  name  helium. ^ 

To  all  these  different  kinds  of  matter  the  properties  of  inertia  and 
weight  are  common;  but  they  are  differentiated  by  many  properties  which 
are  not  common,  such  as  solid,  fluid  or  gaseous  condition,  hardness, 
elasticity,  specific  heat,  heat  and  electric  conductivity,  magnetic  properties, 
color  transparency  or  opacity,  and  so  on.  These  properties  we  shall  take 
up  in  detail  in  their  proper  places. 


(1)  See  Ramsay,  New  Elements,  Nature  Vol.  63,  p.  164. 


26  PROPERTIES     OF    MATTER. 

4:3.  Discontinuity. — One  gas  will  diffuse  through  another;  sugar  or 
salt  will  dissolve  in  and  diffuse  through  water  without  increasing  its  volume; 
platinum  will  absorb  hydrogen;  carbon  monoxide  will  diffuse  through  hot 
iron.  Substances  may  be  reduced  in  volume  by  pressure  or  expanded  by 
heat.  There  must  be  spaces  in  which  the  parts  may  be  pushed  closer  or 
separated.  The  inference  seems  unavoidable  that  matter  is  discontinuous; 
that  is,  it  is  composed  of  parts  so  small  as  to  be  invisible  even  under  the 
microscope,  but  having  interstices  between  them  which  can  hold  particles 
of  other  matter,  in  much  the  same  way  that  a  bushel  measure  filled  with 
apples  may  have  the  interspaces  filled  with  peas  without  any  increase  of 
external  volume. 

4:4:.  Divisibility. — Matter  is  capable  of  division  into  very  small 
parts,  each  of  which  may  retain  many  of  the  properties  of  the  whole.  A 
mere  trace  of  permanganate  of  potassium  will  color  a  quantity  of  water, 
and  the  most  powerful  microscope  will  show  no  discontinuity  of  distribu- 
tion. The  smallest  infusoria  visible  under  a  microscope  appear  to  have 
complete  organs  of  nutrition.  A  small  particle  of  salt  may  be  dissolved  in 
a  barrel  of  water,  and  yet  the  spectroscope  will  show  the  presence  of  some 
of  the  salt  in  every  drop  of  the  water.  A  trace  of  any  sodium  salt  will 
color  a  Bunsen  flame  for  hours.  Quartz  may  be  melted  and  drawn  out  into 
such  fine  fibers  that  they  disappear  under  the  most  powerful  microscope. 
In  all  such  cases  the  subdivision  has  passed  far  beyond  the  direct  perception 
of  our  senses,  but  chemical  phenomena  indicate  that  it  may  actually  pro- 
ceed further.  The  phenomena  of  chemical  combination  in  definite  pro- 
portions indicate,  however,  that  there  is  a  limit,  the  atom,  and  that  the 
atoms  of  each  element  are  alike  in  all  respects.  In  the  case  of  compounds 
a  larger  unit,  the  molecule,  is  assumed,  each  molecule  being  composed  of 
two  or  more  unlike  atoms.  We  shall  find  some  reason  for  believing  that 
in  the  case  of  both  elements  and  compounds  there  are,  particularly  in  the 
case  of  solids  and  liquids,  complex  physical  molecules,  built  up  of  several 
chemical  molecules.  We  shall  also  find  some  reasons  for  at  least  doubting 
whether  the  chemical  atom  is  the  ultimate  unit  of  matter.  Some  recently 
observed  physical  phenomena  (Cathode,  Rontgen,  Becquerel  rays),  seem 
to  indicate  the  existence  of  particles  much  smaller  than  the  hydrogen 
atom.  It  must  be  remembered  that  if  there  be  molecules,  they  are  not 
in  any  way  directly  perceptible.  The  molecular  and  atomic  theory  is  a 
most  useful  one,  which  consistently  ' '  explains  ' '  many  physical  phenomena, 
and  has  been  fruitful  in  leading  to  new  discoveries;  but  after  all  it  is  .but  a 
hypothesis.  No  one  should  accept  it  as  his  creed,  nor  take  too  literally 
the  speculations  regarding  molecular  structure,  mechanism  and  magnitude, 
which  are  so  often  made.  For  convenience  we  may  use  the  term  molecule 
as  indicating  simply  the  smallest  parts  of  substances  which  take  part  in 
physical  processes. 

45.  Molecular  Forces. — WTe  shall  first  consider  the  purely  mechani- 
cal properties  of  matter — those  which  we  assume  to  depend  on  the  size, 
shape>  mass  and  position  of  the  molecules,  and  the  forces  acting  between 
them,  and  the  effects  of  external  forces  upon  molecular  conditions. 

Besides  producing  motion  of  bodies  as  a  whole,  external  forces  may 
cause  strains  in  matter,  or  deformations  of  volume  or  shape  involving 
relative  molecular  motion,  and  work  in  opposition  to  resistance  of  inter- 
molecular  forces. 

46.  States  Of  Matter. — There  are  two  kinds  of  strain — change  of 
volume,  which  depends  upon  the  compressibility  of  the  substance;  change 


STATES    OF    MATTER  —  FLUIDS.  27 

of  shape,  which  depends  upon  its  stiffness  or  rigidity.  We  may  roughly 
divide  matter  into  three  classes  or  states,  depending  upon  the  nature  of 
the  strain  produced  by  an  applied  force.  These  are  solids,  which  are 
almost  incompressible,  and  almost  rigid;  liquids,  which  are  almost  incom- 
pressible, and  not  at  all  rigid;  and  gases,  which  are  compressible  and  not 
at  all  rigid.  Both  liquids  and  gases  have  great  molecular  mobility,  causing 
them  to  flow  or  to  yield  to  the  action  of  the  smallest  applied  forces. 
They  are  called  flidds  in  consequence.  The  distinction  between  liquids 
and  gases  is  that  the  former  have  definite  volumes  and  may  have  free 
surfaces,  the  latter  will  fill  any  vessel  in  which  they  may  be  placed,  and 
usually  have  no  well-marked  free  surface.  (  Some  heavy  vapors  like  iodine 
vapor  may  have  an  indistinct  surface  boundary.) 

In  many  substances  there  appear  to  be  intermolecular  forces  which 
oppose  any  changes  in  the  relative  positions  of  the  molecules,  and  which 
restore  them  to  their  former  positions  if  the  applied  forces  are  not  so 
great  as  to  cause  permanent  rupture.  Such  bodies  are  called  elastic,  and 
the  force  of  restitution  is  a  measure  of  the  elasticity  of  the  substance.  The 
elasticity  is  expressed  in  terms  of  the  force  of  restitution  per  unit  area. 
When  elasticity  is  perfect  this  stress  is  equal  to  the  applied  force  ;  in 
recovering  its  original  condition  the  body  does  as  much  work  as  that  done 
upon  it,  and  there  is  no  loss  in  overcoming  frictional  resistance.  A  well- 
tempered  spring  nearly  fulfills  this  condition.  No  solids  are  perfectly 
elastic  in  any  respect,  however.  All  fluids  are  perfectly  elastic  as  regards 
volume  changes,  but  none  have  elasticity  of  shape.  Some  substances, 
such  as  putty  or  clay,  offer  considerable  resistance  to  forces  tending  to 
change  their  shape,  but  no  force  of  restitution;  the  work  of  the  force  is 
entirely  spent  in  overcoming  internal  friction,  or  viscosity. 

The  various  coefficients  of  elasticity  are  numerically  defined  as  the 
ratio  of  the  force  of  restitution  and  the  strain  produced  by  the  applied 
force. 

(55) 


strain 

FLUIDS. 

47.  Fluids  (including  both  liquids  and  gases)  will  all  yield  in  time 
to  the  action  of  any  distorting  force,  however  small  it  may  be.  All  known 
fluids  offer  some  resistance  to  changes  of  shape,  but  it  breaks  down  with 
time  —  in  some  cases  almost  instantaneously,  in  others  very  gradually. 
This  resistance  is  of  a  frictional  character,  and  seems  to  exist  between  the 
smallest  parts,  or  molecules.  In  the  case  of  fluids  it  is  called  viscosity, 
and  bodies  which  yield  very  slowly  to  distorting  forces  are  said  to  be  very 
viscous.  Molasses  and  tar  are  instances  of  this  kind.  Even  the  lightest 
gases  have  some  viscosity.  Air,  for  example,  offers  a  frictional  resistance 
to  projectiles,  which  increases  very  rapidly  with  increased  velocity. 
Meteors  are  so  heated  by  this  means,  even  in  the  very  rare  upper  atmos- 
phere, that  they  become  incandescent  and  vaporize.  Between  the  least 
viscous  gas,  hydrogen,  and  solids  at  the  other  extreme,  there  is  no  sharp 
dividing  line.  Hydrogen,  air,  carbon  dioxide,  ether,  water,  oil,  molasses, 
shoemaker's  wax,  sealing  wax,  are  examples  of  fluids  in  increasing  order 
of  viscosity.  Many  metals,  such  as  lead,  will  flow  under  the  action  of 
forces  sufficiently  great,  but  an  infinitesimal  force  will  not  cause  this  effect, 
so  that  lead  would  not  be  called  a  fluid  [A.  rod  of  sealing  wax  will  in 


OF  THE 

UNIVERSITY 


28  PROPERTIES     OF    MATTER. 

time  bend  under  the  action  of  very  small  forces;  so  in  a  sense  sealing  wax 
may  be  classed  among  fluids,  although  when  acted  upon  by  a  strong  and 
sudden  force  it  may  exhibit  the  properties  of  a  brittle  solid,  and  appear  to 
be  less  fluid  than  lead.  Heat  in  nearly  all  cases  increases  molecular 
mobility  —  indeed,  is  held  to  be  synonymous  with  it  —  and  thus  changes 
solids  to  liquids,  liquids  to  gases. 

4:8.  Hydrostatics  is  the  branch  of  Mechanics  which  deals  with  the 
equilibrium  of  fluids  at  rest. 

Fluid  Pressure.  When  a  fluid,  either  gaseous  or  liquid,  is  contained 
in  a  vessel,  the  weight  of  the  fluid,  its  elastic  force  of  expansion  (if  a  gas) 
and  forces  transmitted  from  external  sources  will  act  upon  the  walls  of  the 
vessel.  Some  conclusions,  easily  verified  by  experiment,  are  necessary 
consequences  of  the  mobility  of  fluids.  One  of  these  is  that  the  force  of 
a  fluid  at  rest  upon  the  walls  of  the  vessel^  or  upon  any  immersed  surface, 
must  be  everywhere  normal  to  the  surface.  If  there  were  an  uncompen- 
sated  tangential  component,  the  fluid  would  move  in  that  direction,  and 
thus  not  be  at  rest.  With  gravity  alone  acting,  all  liquid  surfaces  are  level; 
when  the  wind  blows  strongly  they  are  normal  to  the  resultant  of  gravita- 
tion and  wind.  If  a  vessel  containing  ether  or  water  is  slightly  tipped, 
readjustment  of  level  takes  place  very  quickly;  if  it  contains  molasses  the 
raised  portion  of  the  liquid  flows  very  slowly  down  hill,  but  in  time  the 
surface  will  become  horizontal. 

It  is  convenient  to  consider  the  force  acting  normally  on  each  unit 
area  rather  than  the  total  force.  Pressure  is  defined  as  normal  force  per 
unit  area.  When  we  speak  of  pressure  at  a  point  we  mean  the  ratio 
between  the  normal  force  acting  on  an  infinitesimal  element  of  surface  and 
the  area  of  that  element. 

49.  Transmission  of  Pressure.  —  Imagine  a  fluid  compressed  by 
a  piston  in  a  cylinder.     If  the  reaction  pressure  in  any  one  region  of  the 
boundary  were  less  than  the  applied  pressure,  the  fluid  would  move  in  that 
direction,  displacing  the  boundary.     It  follows  that  the  applied  pressure 
must  be  transmitted  uniformly  in  all  directions,  and  must  be  the  same  at 
every  point  of  the  surface. 

The  applied  pressure  upon  any  imaginary  element  of  the  fluid  within 
its  volume  must  for  the  same  reason  be  the  same  at  every  point  and  act 
uniformly  in  all  directions. 

50.  Pressure  due  to  Weight  of  the  Fluid.  —  The  pressure  due 
to  an  applied  force  is  the  same  throughout  a  fluid.     This  is  not  true  of  the 
pressure  due  to  the  weight  of  the  fluid.     Consider  any  imaginary  horizon- 
tal surface  within  the  fluid.     Each  unit  area  supports  the  weight  of  the 
column  of  fluid  resting  on  that  area.     This  downward  pressure  is  balanced 
by  an  equal  upward  pressure.     At  every  point  in  the  horizontal  plane  the 
pressure  is  the  same,  but  in  going  downward  it  increases  with  the  depth. 
These  principles  were  first  clearly  recognized  and  stated  by  Pascal. 

In  case  of  a  fluid  of  uniform  density  p  the  pressure  on  a  plane  at  a 
depth  h  is 

(56)  P 


51.  Buoyancy.  —  Any  interior  portion  of  a  fluid  in  equilibrium  must 
be  buoyed  up  by  a  force  exactly  equal  to  its  own  weight,  due  to  the 
upward  pressure  of  the  fluid  beneath  it.  If  we  substitute  for  this  portion  an 
equal  volume  of  any  other  material  of  the  same  density  as  the  fluid,  we 
should  expect  that  to  be  also  in  equilibium  ;  and  experiment  verifies  this 


BUOYANCY SPECIFIC   GRAVITY.  29 

supposition.  In  other  words,  any  submerged  body  is  buoyed  up  by  a 
force  equal  to  the  weight  of  the  water  it  displaces.  If  the  body  is  denser 
than  water,  it  will  sink,  with  an  apparent  weight  equal  to  the  difference 
between  its  own  weight  and  that  of  the  displaced  water.  If  it  is  less  dense, 
it  will  float,  but  still  displace  by  its  submerged  portion  its  own  weight  of 
water.  This  is  known  as  the  principle  of  Archimedes,  and  is  one  of  the 
oldest  laws  of  Physics. 

Archimedes'  principle  is  a  consequence  of  Pascal's  principle.  Consider 
an  imaginary  cube  of  a  fluid  of  sides  a,  in  equilibrium  with  the  same  surround- 
ing fluid.  If  the  upper  surface  be  at  a  depth  h,  below  the  surface,  the 
pressure  per  square  centimeter  on  this  surface  is  Pg/i,  the  total  downward 
pressure  being  a*pgh.  The  lateral  pressures  on  opposite  sides  balance  each 
other.  The  pressure  on  a  horizontal  plane  at  a  depth  h-\-a  below  the  sur- 
face is  pg{h\a~).  Since  this  acts  uniformly  in  all  directions,  the  upward 
pfessure  on  the  bottom  of  the  cube  is  a2pg(/ijra).  The  difference  between 
this  and  the  downward  pressure  gives  as  the  total  buoyant  force,  a?pg — 
but  this  is  equal  to  the  weight  of  the  cube.  If  this  cube  be  replaced 
by  one  of  a  different  density,  the  upward  force  from  without  will  be 
unchanged  and  the  resultant  weight  will  a3(pt — p)g. 

In  the  case  of  two  liquids  in  communicating  vertical  tubes  (U-tubes) 
the  heights  in  each  limb  when  in  equilibrium  depend  only  upon  the  density 
of  the  two  liquids,  not  upon  the  cross  section  or  the  shape  of  the  tube. 

(57)  P.g/1,  =  P2gh2 

This  applies  to  gases  as  well  as  to  liquids.  The  barometer  is  an 
example,  in  which  a  certain  column  of  mercury  in  a  small  tube  is  counterbal- 
anced by  a  column  of  air  in  an  unlimited  space.  If  a  U-tube  with  unequal 
arms  be  held  in  a  cone  of  light  passing  from  a  lens  to  a  screen,  ether 
vapor  poured  into  the  long  arm  will,  by  projection  on  the  screen,  be  seen 
to  flow  out  of  the  short  arm;  if  it  be  poured  into  the  short  arm,  it  will 
simply  flow  over  when  that  arm  is  full,  not  rising  to  the  level  of  the  top 
of  the  long  arm. 

52.  Specific  gravity  is  the  ratio  between  the  weight  of  a  given 
substance  and  that  of  an  equal  volume  of  a  standard  'substance,  usually 
water.  It  is  usually  found  by  an  application  of  Archimedes'  principle. 
A  body  weighs  in  air  W^  in  water  IV2.  By  this  principle  the  weight  of 
an  equal  volume  of  water  is  W^  —  Wz  and 

(58)  s=  T  „**;„.. 


In  the  C.  G.  S.  system  the  specific  gravity  of  a  substance  is  numerically 
equal  to  its  density. 

The  C.  G.  S.  densities  of  some  substances  are  as  follows  : 

Copper  ...................................     8.9 


Water  at  4°  C 1 

Camphor 1 

Pine  wood 0.5 

Potassium 87 

Aluminum..,  2.56 


Mercury  ...................................  13.6 

Gold  .......................................  19.3 

Platinum  .................................  21.5 


Density  varies  with  temperature,  and  in  case  of  solids  with  the  treat- 
ment to  which  subjected  (hammering,  etc.). 

For    historical   discussion    of  laws  of    hydrostatics    see    Mach,   Science    of 
Mechanics,  p.  86. 


80  PROPERTIES    OF    MATTER. 

QUESTIONS  AND  PROBLEMS. 

41.  Show  that  the  total  fluid  pressure  on  the  base  of  a  vessel  depends  solely 
on  its  area  and  the  height  of  the  free  surface  above  it,  not  at  all  on  the  shape  of 
the  vessel. 

42.  Explain  the  hydraulic  press. 

43.  A  cubical  block  of  wood  sinks  to  0.6  of  its  height  in  alcohol  (s.  g.  0.82). 
What  is  its  specific  gravity? 

44.  Given,  a  meter  rod  balanced  on  a  knife-edge,  a  beaker  of  water,  and  a 
stone.     Show  how  to  measure  the  specific  gravity  of  a  body. 

GASES. 

53.  The  capacity  of  indefinite  expansion  possessed  by  gases  indicates 
that  they  possess  much  more  molecular  freedom  than  liquids,  and  for  this 
reason  many  of  their  properties  are  simpler  than  those  of  liquids;  conse- 
quently they  will  be  first  considered. 

Since  gases  are  fluids,  Archimedes'  and  Pascal's  principles  apply  to 
them  without  modification. 

Weight  of  Gases.  Before  the  time  of  Galileo  the  use  of  syringes 
and  pumps  and  many  of  the  phenomena  of  suction  were  known,  but  their 
mode  of  operation  was  not  understood.  The  only  explanation  advanced 
was  that  nature  abhors  a  vacuum.  Galileo  first  began  to  consider  this 
question  when  he  was  told  that  a  certain  pump  with  a  long  suction  pipe 
was  unable  to  raise  water  to  a  height  of  more  than  thirty-three  feet.  He 
had  previously  shown  that  air  has  weight,  by  expelling  some  air  from  a 
large  bottle  by  heating  it  and  noting  the  loss  of  weight,  and  he  seems  to 
have  suspected  some  connection  between  the  weight  of  the  air  and  the 
height  to  which  water  could  be  pumped. 

Torricelli  verified  this  conclusion.  One  of  his  students,  Viviani,  in 
1643,  filled  a  long  glass  tube,  sealed  at  one  end,  with  mercury,  and 
inverted  it  in  a  vessel  of  mercury.  As  Torricelli  had  expected,  the  mer- 
cury column  did  not  entirely  drop,  but  remained  at  a  height  of  about 
one-fourteenth  that  of  the  column  of  water  that  a  pump  can  raise.  Thus 
it  was  shown  that  the  unknown  cause  could  raise  equal  weights  of  different 
liquids,  and  Torricelli  inferred  that  this  cause  must  be  the  counterbal- 
ancing weight  of  the  atmosphere. 

In  1646  the  news  of  Torricelli' s  discovery  came  to  Pascal,  in  France. 
Pascal  likewise  explained  the  result  as  being  due  to  the  weight  of  the 
atmosphere,  and  proved  this  conclusion  in  1648  by  sending  a  barometer 
to  the  top  of  the  Puy  de  Dome.  As  he  expected,  the  height  of  the  mer- 
cury was  less,  on  account  of  the  reduced  weight  of  the  atmosphere  above 
the  open  surface  of  mercury. 

In  1755  Black  discovered  carbon  dioxide,  and  eleven  years  later 
Cavendish  discovered  hydrogen.  Since  then  many  other  gaseous  sub- 
stances have  become  known.  Their  relative  weights  or  densities  may  be 
determined  by  a  refinement  of  Galileo's  method.  Large  globes  may  be 
weighed  when  exhausted  by  an  air  pump,  and  again  when  filled  by  the 
gas  in  question.  The  movements  of  one  gas  in  another  may  be  shown 
visually  by  passing  a  divergent  cone  of  light  through  the  region  occupied 
by  them;  on  a  screen  beyond  the  phenomena  will  be  projected,  being 
made  visible  by  the  different  refractive  powers  of  the  different  gases. 
Ether  vapor  will  be  seen  to  sink  in  air,  hydrogen  to  rise.  (This  is  known 
as  Topler's  schlieren  method.) 


BOYLE'S  LAW — CHARLES'  LAW. 


31 


The  density  of  a  gas  depends  of  course  upon  the  pressure  and  the 
temperature  to  which  it  is  subjected.  The  standard  conditions  are  taken 
to  be  an  atmospheric  pressure  of  76  cm.  of  mercury  and  0°  C.  tempera- 
ture. Under  these  conditions  the  densities  of  some  gases  are  : 


Air  ....................................  0.001293 

Argon  ................................  0.00170 

Carbon  dioxide  ..................  0.00195 

Chlorine  ...................  .  .........  0.00317 


Helium 0.00021 

Hydrogen 0.0000898 

Nitrogen 0.001254 

Oxygen 0.001429 


54.  Boyle's  Law. — About  1662  Robert  Boyle  became  interested 
in  the  experiments  of  Pascal,  which  showed  that  ' '  the  greater  the  weight 
is  that  leans  upon  the  air,  the  more  forcible  is  its  endeavor  of  dilatation. ' ' 
Boyle  undertook  a  careful  investigation  of  the  relation  between  volume 
and  pressure  by  compressing  air  in  the  short  closed  branch  of  a  U-tube 
by  a  column  of  mercury  in  the  long  open  branch.  His  investigations 
went  from  pressure  of  about  one  inch  of  mercury  to  four  atmospheres,  and 
he  found  that  the  volume  varied  very  nearly  inversely  as  the  pressure,  or 


(59) 


pv  =  constant. 


Boyle  also  noted  that  the  volume  and  pressure  were  increased  by  a 
rise  of  temperature,  but  he  did  not  study  this  point  closely. 

In  1676  Mariotte,  of  France,  published  an  account  of  similar  experi- 
ments, but  credit  belongs  by  priority  to  Boyle. 

Boyle's  law  is  a  consequence  of  the  perfect  elasticity  of  gases.  Elas- 
ticity is  numerically  measured  by  the  ratio  of  stress  per  unit  area  to  the 
strain  per  unit  volume;  if  it  is  perfect,  this  ratio  should  be  equal  to  the 
applied  force  per  unit  area.  If  a  gas  is  compressed  by  an  increase  of 
applied  force  from/>0  to/x>  and  the  volume  thereby  reduced  from  v0  to  vlt 
we  have 


(60) 


stress       /  —p0  v    ,  r 

— —  =  — — —  =  /_,  the  applied  force. 

offoi-r*  *?t     ^—  TI  **     x  A    L 


strain 


which  gives 
(61) 


i  vi  =P°  v°~  constant 


55.  Charles'  Law. — Toward  the  end  of  the  eighteenth  century  it 
was  discovered  independently  by  Charles  and  by  Gay-Lussac  that  when 
the  temperature  of  a  gas  changes  it  increases  in  volume  or  in  pressure  or 
both  conjointly  by  the  same  fraction  of  its  volume  at  0  C.  for  each  degree 
change  of  temperature  above  zero  C.  The  value  of  a,  the  coefficient  of 
expansion  or  of  compressibility,  is  J/73  or  0.00367.  The  change  is  an 
arithmetical,  not  a  geometrical  progression,  so  that  at  a  definite  negative 
temperature  of  — 273°  the  product  pv  would  vanish,  provided  the  gas 
retained  its  gaseous  properties ;  as  a  matter  of  fact,  all  gases  become 
liquids  before  that  point  is  reached.  We  cannot  conceive  that  under  any 
circumstances  the  volume  could  become  zero ;  but  the  pressure  may. 

We  may  write 


(62) 


pv  =p.v.  (1  j-  af)  =  ap0v0  (273  ff)= 


where  T  is  the  temperature  measured  from  the  absolute  zero  of  273°  C. , 
and  R  is  the  so-called  gas  constant,  which  is  different  for  different  gases. 


•32  PROPERTIES     OF    MATTER. 

If  p0  and  v0  are  the  pressure  and  the  volume  of  a  unit  mass  of  the  gas 
considered,  the  value  of  the  product  will  be  different  if  a  different  mass  m 
is  employed.  In  general, 

(63)  pv  =  mRT 

where  R  is  the  constant  deduced  from  observations  on  unit  mass. 

56.  Deviations. — It  is  found,  as  in  the  case  of  many  physical  laws, 
that  Boyle's  and  Charles'  laws  are  only  approximately  true,  and  that  wide 
deviations  may  occur  at  high  pressures  and  temperatures.  Regnault; 
Natterer,  and  many  others  have  investigated  these  deviations,  but  the  latest 
and  most  reliable  measurements  are  due  to  Amagat,  now  of  the  Ecole 
Polytechnique  of  Paris.  His  first  experiments  were  made  by  an  extension 
of  Boyle's  method,  the  long  branch  of  his  U-tube  being  a  steel  tube  330 
meters  long  passing  down  the  shaft  of  a  coal  mine  at  St.  Etienne.  Later 
he  used  an  improved  method  by  which  he  could  attain  pressures  of  about 
3000  atmospheres.  The  total  hydrostatic  pressure  due  to  a  column  of 
mercury  acting  on  a  large  piston  A  was  transmitted  through  the  piston 
rod  to  a  smaller  piston  a  which  compressed  the  gas.  This  scheme  is  easily 
seen  to  be  an  inversion  of  the  hydraulic  press.  Instead  of  multiplying 
the  pressure  exerted  on  a  small  area  by  applying  it  to  a  larger  area  by 
means  of  a  liquid  piston,  and  thus  securing  a  larger  total  pressure,  the 
total  pressure  secured  in  such  a  manner  is  by  means  of  a  solid  piston  con- 
centrated on  a  small  area.  If  A  be  the  area  of  the  large  piston,  a  that  of 
the  small  piston,  h  the  height  of  the  mercury  column  which  produces  the 
pressure,  and  p  the  density  of  mercury, 

(64)  P  = 

is  the  pressure  applied  to  the  gas  in  the  small  cylinder. 

Amagat  lubricated  the  pistons  with  very  viscous  liquids,  such  as  castor 
oil  and  molasses,  which  effectively  prevented  any  leakage  of  mercury  or 
gas  around  the  pistons. 

Some  of  his  results  for  different  gases  are  summarized  below.  The 
product pv  at  atmospheric  pressure  and  0°  C.  is  taken  as  unity;  if  Boyle's 
law  were  exactly  obeyed,  this  product  should  remain  constant  with  con- 
stant temperature.  The  curves  drawn  with  the  product  pv  as  ordinates 
and/  as  abscissae,  show  deviations  from  the  law.  Each  curve  should  be 
a  horizontal  line  if  the  law  were  correct. 

Value  ofpv  (p  v  =1). 
Pressur-  in  Atmos.  Oxygen.      Hydrogen.      Nitrogen.    Carbon  Dioxide. 

1.0000 
0.1050 
0.2020 
0.3850 
0.5595 
0  8905 
1.6560 


1  

1  0000 

1  0000 

1  0000 

50  

100  

9055 

1  0690 

9910 

200  

9140 

1  1380 

1  0390 

300  

9625 

1  2090 

1  1360 

500  

1  1570 

1  QfSfir; 

i  QUOO 

1000   

1  7360 

1  79^0 

9  0700 

2000  

2  8160 

2  38QO 

Q  3270 

2800... 

3  6176 

2  8686 

4  2700 

In  the  case  of  hydrogen,  the  value  of  pv  uniformly  increases  as  the 
pressure  rises,  or  the  gas  is  not  so  compressible  as  required  by  Boyle's 
law.  In  the  case  of  oxygen,  nitrogen,  and  the  heavier  gases,  such  as 
carbon  dioxide  and  ethylene,  pv  diminishes  to  a  minimum  and  then  rapidly 
increases  with  increasing  pressure,  being  at  first  more  then  less  com- 


DEVIATIONS D ALTON'S    LAW.  33 

pressible  than  the  law  would  require.  This  suggests  that  for  all  the  gases 
except  hydrogen  there  is  up  to  a  certain  point  a  slight  attraction  between 
the  molecules  which  helps  condensation;  then  a  crowding  together  which 
resists  it.  As  the  crowding  increases  until  the  point  of  liquefaction  is 
approached,  the  deviation  from  Boyle's  law  becomes  very  great.  The 
easily-liquefied  gases,  such  as  carbon  and  sulphur  dioxide,  chlorine, 
ethylene,  etc.,  show  marked  deviations.  In  the  case  of  hydrogen  there  is 
always  the  internal  resistance  to  compression  with  little  or  no  molecular 
attraction.  In  a  gas  which  strictly  obeys  Boyle's  law  there  could  be  no 
molecular  attraction  or  repulsion;  the  pressure  would  depend  solely  on 
the  number  of  molecules  within  a  given  volume,  without  regard  to  any 
forces  between  them. 

It  is  seen  that  Boyle's  law  is  more  nearly  true  at  high  temperatures; 
it  is  believed  to  hold  very  closely  for  low  pressures,  but  it  has  been  found 
hard  to  verify  this  conclusion  because  of  the  difficulties  of  measuring 
small  pressures  accurately. 

It  would  seem  natural,  as  indicated  above,  to  explain  deviations  from 
Boyle's  law  as  being  due  to  two  causes,  (1)  intermolecular  forces  of 
attraction,  helping  the  applied  pressure  to  condense  the  gas;  (2)  the 
actual  volume  occupied  by  the  molecules,  or  the  volume  they  would  fill 
were  there  no  free  space  between  them.  Attempts  have  been  made  to 
generalize  Boyle's  law  by  taking  account  of  these  factors.  One  of  the 
most  successful  is  that  of  van  der  Waals,  who  in  1873  proposed  the 
formula 

(65)  ^+^)  (v-b}  =  mRT, 

which  very  closely  conforms  to  observation  for  all  gases  and  vapors. 
The  term  a  /  v2  takes  account  of  the  effect  of  molecular  attractions;  the 
term  b  expresses  the  molecular  volume,  or  the  part  of  space  actually 
occupied  by  the  molecules.  For  air  a  =  0.0037;  £  =  0.0026.  For 
carbon  dioxide  a  =  0.01 15;  *=-0.003.^U  ^*  /**,  W.  *  M>  *f  I *+} 

57.  Dalton's  Law. — In  a  mixture  of  several  gases  having  no  chem- 
ical action  on  each  other  the  total  pressure  is  equal  to  the  sum  of  the 
partial  pressures  produced  by  each  separately. 

(66) 

(For  original  papers  of  Boyle  and  Amagat  see  Barus,  the  Laws  of  Gases;  also, 
for  results,  Tait,  Properties  of  Matter.) 

THE   ATMOSPHERE. 

58.  The  pressure  at  any  point  in  the  earth's  atmosphere  is  due  mainly 
to  the  weight  per   unit  area  of  the  air  above  that  point,  although  the 
pressure    may    be   somewhat    modified    by   atmospheric   currents.     The 
density  diminishes  very  rapidly,  therefore,  in  going  upward. 

The  composition  of  the  atmosphere  at  sea  level  is  approximately  as 
follows:  Nitrogen,  .78;  oxygen,  .21;  argon,  .01;  varying  traces  of  car- 
bon dioxide,  ammonia,  and  water  vapor;  very  small  traces  of  the  gases 
neon  and  krypton.  These  properties  are  subject  to  slight  local  variations. 

Since  oxygen  is  somewhat  heavier  than  nitrogen,  there  is  a  slight 
tendency  for  it  to  accumulate  more  at  the  bottom  than  at  the  top  of  the 


34  PROPERTIES     OF     MATTER. 

atmosphere.  In  ascending,  therefore,  the  proportion  of  oxygen  becomes 
slightly  less. 

At  the  sea  level  the  atmospheric  pressure  is  on  the  average  equal  to 
the  weight  of  a  column  of  mercury  76  cm.  high  and  of  1  square  cm.  cross 
section.  This  is  equivalent  to  a  pressure  of  about  1,013,000  dynes  per 
square  cm.,  or  14.7  pounds  weight  per  square  inch. 

The  law  of  diminution  of  pressure  in  ascending  above  sea  level  is 
readily  calculated  by -the  application  of  Boyle's  law.  The  calculated 
percentages,  at  several  different  heights,  of  the  principal  constituents,  and 
the  pressure  of  the  mixture,  are  given  below: 

Height  in  Relative  Percentage. 

Meters.  Oxygen.  Nitrogen.         Total  Pressure. 

0  21  78.96  760      mm. 

1,000  20.71  79.25  670.7 

10,000  18.35  81.63  218 

20,000  15.92  84.07              62.8 

40,000  11.54  88.46                5.2 

60,000  8.89  91.11                0.4 

The  quantity  of  oxygen  becomes  not  only  absolutely  but  relatively 
less  in  ascending. 

Theoretically  there  is  no  limit  to  the  height  of  the  atmosphere  ;  but  at 
an  infinite  distance  the  density  would  be  infinitely  small.  At  a  height  of 
several  hundred  miles  the  density  is  sufficient  to  raise  meteors  to  incan- 
descence by  the  frictional  resistance  offered  to  their  very  rapid  motion. 
The  greatest  altitude  was  probably  reached  by  Glaisher  in  a  balloon  ascent 
in  1862.  He  went  to  a  height  of  nearly  seven  miles,  his  barometer  falling 
to  a  height  of  less  than  nine  inches. 

The  usual  method  of  indicating  and  measuring  the  atmospheric  pres- 
sure is  by  means  of  the  barometer,  which  was  devised  by  Torricelli  in 
1643.  Shortly  afterward  Otto  von  Guericke  invented  a  mechanical  air 
pump  and  showed  the  atmospheric  pressure  by  the  classical  experiment  of 
the  Magdeburg  hemispheres.  These  hollow  metallic  hemispheres,  fitting 
closely  at  their  edges,  were  held  together  with  great  force  when  the  air 
between  them  was  exhausted. 

The  distinction  between  the  weight  and  the  elastic  pressure  of  a  gas 
must  be  carefully  borne  in  mind.  Any  portion  of  air  at  sea  level  is  com- 
pressed by  the  weight  of  the  atmosphere  above  and  exerts  an  equal  and 
opposite  elastic  force  of  one  atmosphere,  whether  the  actual  mass  of  the 
portion  be  great  or  small.  If  we  introduce  in  a  barometer  tube  enough 
air  to  depress  the  mercury  to  one-half  its  former  height,  it  by  no  mea'ns 
follows  that  we  have  introduced  seven  and  a  third  pounds  of  air. 

59.  Barometers. — The  simplest  type  of  barometer  is  the  siphon,  in 
which  the  pressure  of  the  atmosphere  is  equal  to  the  weight  of  the  column 
of  mercury  between  the  levels  h^  and  h2. 

The  Fortin  type  is  ordinarily  used  in  scientific  work.  The  bottom  of 
the  mercury  reservoir  is  of  leather,  and  may  be  raised  or  lowered  by  a 
screw,  in  order  to  keep  the  surface  at  a  constant  level. 

In  the  aneroid  barometer,  changes  in  pressure  are  registered  by  the 
motion  of  an  index  connected  with  the  elastic  corrugated  metal  cover  of  a 
partially  exhausted  box. 

Various  forms  of  self-registering  barometers  are  used  in  meteorological 
work. 


OF  THE 


X^  r*        OF 

BAROMETERS.  35 

60.  In  order  that  observations  made  at  different  times  and  places  may 
be  comparable,  they  must  be  reduced  to  standard  conditions,  and  expressed 
in  dynes.  The  standard  condition  of  the  barometer  is  a  height  of  76  cm. 
of  mercury  at  a  temperature  of  0°  C.,  at  sea  level,  in  latitude  45°.  On 
account  of  the  variations  of  g,  the  same  height  will  not  indicate  the  same 
pressure  in  dynes  in  any  other  latitude. 

Four  corrections  have  to  be  applied  to  a  barometric  reading,  in  order 
to  determine  the  absolute  pressure  in  dynes,  on  account  of — 

(#)  Capillarity.  On  account  of  surface  tension,  the  mercury  is  de- 
pressed in  a  glass  tube.  This  depression  may  be  quite  large  in  small 
tubes,  but  for  a  tube  of  1  cm.  diameter  it  is  only  about  0.002,  or  neg- 
ligibly small. 

(^)     Change  of  the  density  of  mercury  with  the  temperature,  and — 

(c]  Change  of  the  length  of  scale  with  the  temperature.  These  two 
corrections  may  be  considered  together.  If  r  is  the  scale  reading,  kQ 
the  reduced  height,  and  c  the  cubical  coefficient  of  expansion  of  mercury, 


If  the  scale  has  a  linear  coefficient  of  expansion  <z,  the  true  height 
in  cm.  is 

h  =  r  (1  +  af) 
from  which 

(67)  A.  =  r 


(d)     The  absolute  pressure  in  dynes  is  given  by 


which  depends  on  the  local  value  of  g. 

61.  Measurement  of  Altitudes  by  Barometer.  Imagine  the  atmos- 
phere divided  in  horizontal  strata  of  thickness  yt  so  small  that  there  is  no 
sensible  change  of  density  within  one  stratum,  and  number  them  from 
0  to  n.  Then 


- 

from  which 

j    =  l  +P~gy  = 
j-9    =1  +jgy  = 

p~  =  1  +  P^jy- 

since   by  Boyle's  law  the  ratio  p  jp  is  constant  for  constant  temperature. 
Multiplying  together  the  left-hand  and  the  right-hand  members,  we  have 

A  =  CB    • 

A 

Passing  to  natural  logarithms, 


36  PROPERTIES    OF    MATTER. 

But  H=ny  =  -^-c  log^  =  ^  log  £ 

if  we  put  y  equal  to  1  cm.  and  remember  that  natural  log  (1  +  kg)  =  kg 
when  this  is  a  very  small  quantity. 

Substituting  h^h^  (barometric  heights)  for  pojp*  and  reducing  to  com- 
mon logarithms,  we  have  -finally 


(68)  H=  log  ~  =  18420  log  ~  meters 

f&$£  '^TL  ftfL 

An  exact  expression  would  involve  use  of  the  local  value  of  g  and 
corrections  for  temperature  and  water  vapor  at  the  two  stations. 

In  very  exact  calculations  of  the  density  of  the  atmosphere  the  amount 
of  water  vapor  present  must  be  found  and  a  correction  made,  on  account 
of  the  fact  that  water  vapor  is  lighter  than  air  when  under  the  same 
pressure. 

There  are  regular  daily,  monthly,  and  yearly  variations  of  the  baromet- 
ric height,  principally  the  direct  or  indirect  effects  of  temperature.  There 
are  also  accidental  and  irregular  variations  depending  on  wind  and 
weather.  In  the  tropics  the  accidental  variations  are  small,  rarely  exceed- 
ing a  few  millimeters,  and  the  daily  variations  are  almost  as  regular  as 
clockwork,  the  height  being  a  maximum  at  about  10  A.  M.  and  9  P.  M. 
and  a  minimum  at  about  4  p.  M.  and  4  A.  M.  In  the  temperate  regions 
the  regular  variations  are  almost  masked  by  the  accidental  ones,  these 
variations  amounting  to  as  much  as  several  inches  in  high  latitudes. 

There  are  some  indications  of  exceedingly  small  tidal  variations  in 
atmospheric  pressure,  due  to  the  position  of  the  moon. 

62.  Cyclones.  —  Local  high  temperature  or  other  causes  may  cause 
a  barometric  depression  or  '  '  low  "  at  a  given  place.     The  light  air  rises 
and  heavier  air  comes  in  laterally  to  take  its  place.     The  inertia  of  this 
air  combined  with  the  rotation  of  the  earth  causes  a  deflection  to  the 
right,   as  in  the  case  of  projectiles,  and  this  causes  the  air  to  circulate 
spirally  around  the  low  in  the  direction  opposite  to  the  hands  of  a  clock 
in  the  northern  hemisphere  and  in  the  reverse  direction  in  the  southern 
hemisphere.     The  low  with  the  accompanying   cyclone   usually    travels 
across  country,  generally  from  a  northwest  to  a  southeast  direction  in  the 
United  States.     More  restricted  local  disturbances  of  a  similar  character 
give  rise  to  tornadoes  and  whirlwinds,  and  on  the  ocean  to  water-spouts. 

63.  Manometers.  —  The  barometer  is  a  special  form  of  manometer, 
or  instrument  to  measure  fluid  pressures.      For  pressures  not  differing 
greatly  from  that  of  the  atmosphere,  mercury  in  an  ordinary  U-tube  is 
used,  one  branch  communicating  with  the  gas,  another  with  the  atmosphere. 

For  low  pressures,  vacuum  gauges  are  used.  The  closed  end  of  a 
U-tube  is  completely  filled  with  mercury.  As  the  pressure  on  the  open 
end  diminishes,  the  mercury  will  fall  in  the  closed  end.  For  very  low 
pressures  instruments  like  the  McLeod  gauge  are  used.  A  quantity  of 
the  highly  rarefied  gas  is  first  reduced  to  a  known  fraction  of  its  original 
volume,  its  pressure  then  measured,  and  the  original  pressure  calculated 
by  Boyle's  law. 

For  high  pressures,  air  is  compressed  in  a  closed  tube  by  a  column  of 
mercury  in  a  U-tube,  the  other  surface  of  which  is  exposed  to  the  pressure, 
and  this  pressure  calculated  by  Boyle's  law. 


AIR    PUMPS — MERCURY    PUMPS.  37 

Applications  of  atmospheric  pressure  are  found  in  the  pipette,  the 
siphon,  the  lift-pump,  and  the  force-pump. 

References — Travers,  Experimental  Study  of  Gases;  Ramsay,  Gases  of  the 
Atmosphere;  Greeley,  American  Weather;  Ferrel,  Treatise  on  Winds;  Text-books 
on  Meteorology  by  Russell,  Davis,  Waldo,  Abereromby,  and  Scott. 

QUESTIONS   AND    PROBLEMS. 

45.  Calculate  the  gas  constant  R  for  hydrogen,  oxygen,  nitrogen,  and  carbon 
dioxide.     What  is  the  relation  between  these  different  values  and  the  densities  of 
the  gases  ? 

46.  An  inverted  liter  bottle  filled  with  air  is  sunk  20  feet  under  water.     To 
what  volume  is  the  air  reduced  ? 

47.  About  how  much  would  a  barometer  fall  on  being  carried  from  a  ferry 
boat  to  the  top  of  Grizzly  Peak  (1800  feet)  ? 

64: .  Air  Pumps. — There  are  various  ways  of  removing  air  from  a 
closed  space.  One  of  the  simplest  methods  is  by  the  depression  of  the 
mercury  column  in  a  long  barometer  tube.  If  the  mercury  is  perfectly 
dry  and  free  from  air,  this  space  will  be  a  perfect  vacuum  except  for  the 
presence  of  a  small  quantity  of  mercury  vapor. 

Another  simple  method  is  by  displacing  the  air  by  some  vapor  which 
can  afterward  be  condensed.  This  principle  is  applied  in  making  ther- 
mometers, the  mercury  being  boiled  to  expel  the  air,  and  the  tube  sealed 
while  filled  with  the  mercury  vapor. 

The  oldest  form  of  air  pump  is  the  piston  and  cylinder  pump  invented 
about  1650  by  Otto  von  Guericke,  and  subseqently  improved  by  Boyle 
and  Papin.  In  a  modern  double-acting  pump  the  valves  communicating 
with  the  vessel  to  be  exhausted  are  alternately  closed  while  the  piston  is 
compressing,  and  open  while  the  piston  is  rarefying  the  air  in  the  cylinder. 
The  valves  communicating  with  the  atmosphere  are  open  while  the  others 
are  closed,  permitting  the  escape  of  the  compressed  air. 

If  V  is  the  volume  of  the  receiver  and  its  connections,  and  v  the 
volume  of  the  cylinder,  at  each  stroke  of  the  pump  the  volume  v  is 
expelled,  and  the  pressure  is  reduced  in  the  ratio  VI  V-\- v.  At  the  end 
of  the  nth  stroke  the  pressure  is  reduced  to 

y°  P 

(69)  />. 


An  infinite  number  of  strokes  would  thus  be  necessary  to  secure  a  perfect 
vacuum.  Practically  a  pressure  of  a  few  millimeters  of  mercury  is  the 
lowest  usually  attainable,  as  the  valves  cease  to  act  when  the  pressure 
becomes  too  small  to  move  them. 

65.  Mercury  Pumps  are  much  more  efficient.  There  are  two 
typical  forms,  the  Geissler  and  the  Sprengel.  The  Topler-Hagen  is  an 
improved  form  of  the  Geissler  pump,  working  without  any  stop-cocks. 
When  the  reservoir  R  is  raised,  the  mercury  rises  toward  the  bulb  B, 
cuts  off  communication  with  the  space  ,S  which  is  to  be  exhausted,  and 
drives  the  imprisoned  air  out  through  the  capillary  fall  tube  T.  On  lower- 
ing the  reservoir  a  Torricellean  vacuum  is  left  behind  the  mercury,  air 
rushes  in  from  S,  and  the  operation  is  repeated,  one  bulb  full  of  air  being 
expelled  at  each  stroke.  There  is  no  definite  limit  to  the  possible 
exhaustion,  except  the  almost  unavoidable  presence  of  small  traces  of 
water-vapor,  air,  and  other  gases  which  become  condensed  on  the  glass 
walls  of  the  pump,  and  gradually  escape  into  the  vacuum. 


38  PROPERTIES    OF    MATTER. 

The  Sprengel  pump  acts  on  an  entirely  different  principle.  If  a  jet  of 
mercury  moves  past  the  mouth  of  a  tube  communicating-  with  S,  and  if 
the  point  P  be  at  a  height  greater  than  that  of  the  barometric  column  above 
the  mercury  surface  at  the  bottom  of  the  fall  tube,  it  is  evident  that  the 
hydrostatic  pressure  in  the  mercury  at  P  will  be  zero.  The  weight  of  the 
atmosphere  is  unable  to  sustain  the  weight  of  the  mercury  column  below 
P,  and  there  is  a  tendency  for  the  column  to  break,  as  the  lower  parts  are 
moving  faster  than  the  upper  parts,  which  have  just  begun  to  fall.  Air  is 
consequently  sucked  in  laterally  and  carried  down  with  the  mercury.  The 
same  effects  would  be  produced  if  the  fall  tube  were  much  shorter,  and 
the  mercury  driven  through  the  nozzle  under  the  action  of  a  pressure 
greater  than  that  of  the  atmosphere.  This  is  the  case  with  the  Bunsen  jet 
or  filter  pump,  which  is  short,  but  operated  by  water  under  high  pressure. 
The  action  of  such  pumps  may  seem  at  first  sight  paradoxical;  but  we 
must  remember  that  while  the  jet  is  produced  by  high  pressures,  this 
pressure  does  not  exist  in  the  freely  falling  jet  itself.  As  a  matter  of  fact, 
each  part  of  the  jet,  in  addition  to  the  momentum  which  enables  it  to  move 
against  atmospheric  pressure,  is  subject  to  the  acceleration  of  gravity,  so 
that  the  lower  parts  tend  to  separate  from  the  upper  parts,  thus  creating  a 
tension,  or  negative  pressure.  This  is  a  special  example  of  the  general 
law  of  reduction  of  pressure  in  a  moving  fluid,  to  be  considered  later. 
The  Sprengel  pump  is  even  more  efficient  than  the  Geissler  pump  and  its 
modifications. 

GASES  IN  MOTION. 

66.  Unbalanced  forces  produce  motion  in  matter  ;  conversely,  from 
motion  of  matter  we  may  infer  the  existence  of  unbalanced  forces.  When- 
ever we  see  fluids  in  motion  Pascal'  s  principle  cannot  hold  ;  the  pressures 
on  all  sides  of  a  moving  element  cannot  be  the  same. 

67-  Efflux  of  Gases.  —  Consider  the  flow  of  a  fluid  through  a  small 
opening  a  in  a  very  thin  wall,  the  pressures  on  the  two  sides  being  /x  and 
p^  and  the  work  done  in  driving  through  this  opening  a  small  cylinder 
of  the  fluid  of  cross  section  a  and  length  /.  This  work  is  evidently 

(70)  w=Fl=(p-p^al=( 


where  v  is  the  volume  of  the  cylinder. 

If  all  of  this  work  is  expended  in  producing  kinetic  energy, 

(P—P*}V  =  \ms*  or  p-p2  =  &S* 

(71)  s=  |2(A-A) 

y   ' 

In  the  case  of  liquids  this  expression  (known  as  Torricelli's  law)  is  a  very 
close  approximation,  as  there  is  only  a  slight  loss  of  energy  due  to  friction, 
if  the  wall  is  thin.  In  the  case  of  gases,  there  is  not  only  their  frictional 
resistance,  but  also  loss  of  energy  in  expanding  against  the  exter- 
nal pressure.  This  changes  the  numerical  value  of  the  expression  but 
still  leaves  the  speed  of  efflux  of  different  gases  inversely  proportional 
to  the  square  root  of  the  density  of  the  gas.  This  principle  has  been  used 
by  Bunsen  in  comparing  the  densities  of  gases. 

To  the  extent  that  the  above  formula  is  true  we  see  that  since  p  varies 
as  /  the  velocity  of  efflux  of  a  gas  is  the  same  at  all  pressures.  For 
hydrogen  escaping  into  a  vacuum  under  a  pressure  of  one  atmosphere, 


TRANSPIRATION CHANGES  OF  PRESSURE  DUE  TO  MOTION.  39 

,$•=0.000089  and /=!, 013,000  dynes;  consequently  s  is  about  1500  meters 
per  second;  the  speed  of  air  is  a  little  more  than  one  fourth  of  this. 

68.  Transpiration  is  the  name  applied  to  the  passage  of  gases 
under  a  difference  of  pressure  through  long  capillary  tubes.  The  gas  un- 
usually condenses  upon  or  ' '  wets  ' '  the  walls  of  the  tube,  hence  the  flow 
is  impeded  by  viscosity,  or  the  friction  of  the  gas  upon  itself.  The  rate 
of  flow  no  longer  obeys  any  simple  law.  Below  are  given  the  relative 
rates  of  transpiration  of  some  gases. 

CO2 1,376 


O 1 

N 1,150 


NH3 2 


H  ............................................  2,26 

In  general  the  lighter  gases  transpire  more  rapidly,  but  there  are  excep- 
tions, as  seen  by  comparing  CO2  and  NHy 

69.  Changes  of  Pressure  due  to  Motion.  —  If  there  is  in  any 
fluid  a  positive  acceleration  in  a  given  direction,  there  must  also  be  a  tend- 
ency for  the  parts  going  in  that  direction  to  separate  from  the  parts 
behind  them,  and  to  lag  behind  the  parts  in  front,  thus  diminishing  the 
pressure  between  them;  and  conversely  a  retardation  involves  a  crowding 
together  and  consequent  increase  of  pressure.  This  point  will  be  made 
clear  by  considering  an  analogous  case.  If  a  weight  rests  on  a  board,  it 
exerts  a  certain  pressure  on  the  latter;  if  the  two  are  allowed  to  fall 
freely  while  in  contact  there  is  no  pressure  whatever  between  them;  the 
weight  simply  moves  continuously  into  the  space  formerly  occupied  by  the 
board,  without  pushing  it;  if  the  board  is  not  allowed  to  fall  freely,  but  is 
attached  to  a  cord  moving  with  friction  over  a  pulley,  thus  reducing  its 
acceleration  to  #t,  there  will  be  more  or  less  pressure  depending  on  the 
relative  acceleration,  but  it  will  not  be  equal  to  the  total  weight.  Again, 
imagine  a  heavy  cylinder  with  a  tightly-fitting  piston  moving  in  it  without 
friction.  If  the  cylinder  be  held  vertically  so  that  the  piston  starts  to  fall  out 
of  it,  and  the  cylinder  be  allowed  to  fall  itself  an  instant  later,  the  piston 
having  started  first,  will  at  each  instant  be  falling  faster  than  the  cylinder, 
and  will  thus  increase  the  volume  and  diminish  the  pressure  of  the  air 
contained  in  the  latter. 

In  the  case  of  incompressible  fluids  it  is  possible  to  calculate  a  fairly 
exact  relation  between  velocity  and  pressure,  neglecting  the  effects  of 
friction,  which  are  small  except  in  the  case  of  flow  through  small  pipes, 
such  as  capillary  tubes,  in  which  viscosity  plays  a  large  part.  Consider 
any  small  parallelepipedon  of  fluid  of  cross  sections  and  length  b  with  pres- 
sure /,  and  p2  acting  on  its  two  ends.  If  the  element  moves,  these  pres- 
sures must  be  different,  and  work  is  done  on  the  element  by  a  force  per  unit 
area  equal  to  the  difference  of  pressure.  If  we  neglect  losses  by  friction 
all  this  work  is  transferred  into  kinetic  energy  (or  conversely  if  the 
momentum  of  the  element  is  carrying  it  against  a  higher  pressure);  hence 


(72) 
From  this  equation  we  see  that 

(73)  /,+  \psr2  =p 

This  equation  shows  that  wherever  the  speed  of  a  fluid  is  greatest  the 
hydrostatic  pressure  is  least,  or  conversely;  and  the  sum  of  the  pressure  at 
a  point  and  the  kinetic  energy  of  the  unit  volume  embracing  that  point  is 
a  constant.  This  law,  known  as  Bernouillis  law,  is  only  qualitatively 
true  in  the  case  of  compressible  fluids  like  gases. 


40  PROPERTIES     OF    MATTER. 

If  a  jet  of  fluid  issues  from  a  reservoir  under  atmospheric  pressure 
(£=1,014,000  dynes)- 

(74)  A  =/0  -  fas  =  0  when  J= J^. 

\   P 

For  this  speed,  the  pressure  within  the  jet  is  zero;  for  greater  speeds,  it  is 
negative  or  the  parts  will  tend  to  fly  asunder,  causing  suction  laterally  on 
any  surrounding  fluid  at  rest. 

There  are  many  examples  of  this  principle,  both  in  liquids  and  gases. 
The  Sprengel  pump  is  one  illustration.  If  a  jet  of  liquid  or  air  be  driven 
from  a  tube  through  a  funnel-shaped  expansion,  light  objects  may  be 
sucked  up  into  the  funnel.  A  light  sphere  tends  to  rest  against  an  up- 
ward jet  of  water.  Air  blown  through  a  horizontal  tube  will  suffer  a  re- 
duction of  pressure  on  expanding  into  a  larger  tube,  and  the  reduction  of 
pressure  will  raise  a  liquid  in  a  connected  vertical  tube  T.  If  a  jet  of  air  be 
blown  against  a  metal  plate  near  a  flame  the  flame  will  be  sucked  toward 
the  plate.  A  downward  jet  of  air  will  pick  up  a  card.  Winds  in  general 
reduce  barometric  pressure.  A  condition  for  success  in  all  such  experi- 
ments is  that  the  speed  of  the  moving  fluid  must  be  sufficiently  great  to 
maintain  itself  against  any  obstructions,  such  as  opposing  atmospheric  pres- 
sure, and  flow  in  steady  stream  lines.  A  jet  of  air  or  water  fulfills  these 
conditions,  the  momentum  of  the  fluid  keeping  up  a  steady,  state  of  motion. 
Other  examples  will  be  given  in  discussing  the  motion  of  liquids. 

70.  Diffusion. — The  cases  of  fluid  motion  so  far  considered  are  mass 
motions,  depending  on  differences  of  pressure  on  the  two  sides  of  any  ele- 
mentary mass  considered;  if  any  very  small  element  could  be  frozen  solid 
the  resulting  motion  would  be  the  same.  In  the  case  of  all  fluids  which 
are  capable  of  mixture,  however,  (excluding  such  cases  as  oil  and  water  in 
contact),  we  observe  that  there  are  relative  motions  not  dependent  upon 
differences  of  pressure,  and  which  appears  to  be  molecular,  not  mass, 
effects. 

Dalton  showed  that  if  a  vessel  filled  with  one  gas  be  placed  in  commu- 
nication with  another  vessel  filled  with  a  different  gas,  the  two  would  in 
time  be  found  to  be  uniformly  mixed,  no  matter  how  small  the  channel  of 
communication;  this  is  true  even  if  the  lighter  gas  be  placed  above 
the  heavier.  Such  experiments  illustrate  the  tendency  of  every  gas  to  uni- 
formly fill  all  the  space  open  to  it,  without  regard  to  whatever  other  gas 
may  be  present.  This  process  is  called  diffusion.  The  presence  of 
another  gas  will  retard  the  rate  at  which  a  gas  diffuses,  but  it  in  no  way 
interferes  with  the  final  equilibrium  of  the  gas  with  itself.  In  a  sense  any  gas 
is  a  vacuum  to  any  other  gas.  The  attraction  of  gravitation  is  some  check 
on  indefinite  diffusion,  as  shown  by  the  atmosphere.  Heavier  gases  col- 
lect below  the  lighter,  as  in  the  case  of  "fire  damp"  in  mines  and  wells. 
It  follows  that  Pascal's  principle  must  apply  to  each  gas  in  a  mixture  sepa- 
rately, and  it  can  only  be  in  equilibrium  when  its  pressure  is  the  same  in 
all  directions,  or  its  density  the  same  at  all  points  in  a  horizontal  plane.  If 
we  imagine  the  state  of  a  rarefied  gas,  however,  as  composed  of  isolated 
molecules  with  large  spaces  between  them,  it  is  difficult  to  picture  to  our- 
selves anything  like  a  uniform  hydrostatic  pressure  in  a  gas — such  a  pres- 
sure as  that  between  two  surfaces  everywhere  in  contact.  The  phenomena 
of  diffusion  suggest  very  strongly  that  motion  has  something  to  do  with 
what  we  call  the  pressure  of  a  gas,  and  we  shall  presently  find  that  all  the 
phenomena  of  gases  can  be  explained  very  simply  and  very  satisfactorily  as 


DIFFUSION  -  ABSORPTION  —  OCCLUSION.  41 

a  result  of  motion  alone,  without  any  sort  of  contact  or  pressure  between 
neighboring  molecules  except  that  resulting  from  momentary  impacts. 
This  is  called  the  kinetic  theory  of  gases. 

71.  Diffusion   Through   Porous    Walls—  Atmolysis.—  If  two 

vessels  containing  different  gases  be  separated  by  a  porous  partition,  say  of 
unglazed  earthenware,  it  will  be  found  after  a  time,  even  if  both  gases  be 
originally  at  the  same  pressure,  that  they  will  become  uniformly  mixed. 
During  the  process  of  diffusion  the  equilibrium  of  pressure  will  be  for  a 
time  disturbed,  the  lighter  gas  moving  more  rapidly  than  the  heavier. 
This  may  be  shown  as  follows:  A  glass  tube  is  sealed  into  a  porous  cup 
and  the  whole  inverted  so  that  the  end  of  the  glass  tube  is  submerged  in 
water.  Surround  the  cup  by  an  atmosphere  of  hydrogen  or  coal  gas,  and 
the  pressure  within  the  cup  will  rise,  as  shown  by  the  escape  of  bubbles. 
After  a  time  renew  the  original  surrounding  atmosphere.  The  pressure 
will  now  diminish,  as  shown  by  the  rise  of  water  in  the  tube.  The  coal 
gas  gets  in  at  first  more  rapidly  than  the  air  can  get  out;  afterward  it  gets 
out  more  rapidly  than  the  air  can  get  in.  With  carbon  dioxide  the  con- 
verse is  true.  Graham  showed  that  the  rates  of  interdiffusion  of  two  gases 
are  inversely  as  the  square  root  of  their  densities.  We  may  infer  from 
such  experiments  that  any  gas  is  always  diffusing  within  itself,  or  that  the 
molecules  are  always  in  motion.  This  method  enables  us  to  sift  one  gas 
from  another  (atmolysis). 

We  have  here  another  illustration  of  Dalton's  law.  Each  gas  strives 
to  secure  equilibrium  for  itself  in  all  the  available  space,  without  regard  to 
any  other  gas  that  may  be  present 

72.  Absorption  of  Gases  by  Solids  and  Liquids.  —  Boyle's  law 
and  the  phenomenon  of  diffusion  indicate  that  the  attractive  forces  between 
the  molecules  of  a  gas  or  of  different  gases  are  exceedingly  small.     This 
is  apparently  not  the  case  with  the  attractive  forces  between  gases  and 
solids  or  liquids.     Water  will  absorb  or  dissolve  more  or  less  of  any  gas 
in  contact  with  it.     Some  gases  are  absorbed  but  slightly;  for  example, 
oxygen  and  nitrogen;  others  are  absorbed  in  large  quantities;  for  example, 
ammonia  gas  and  hydrochloric  acid  gas.      It  is  found  that  the  quantity  of 
a  gas  absorbed  by  water  or  other  fluids  is  directly  proportional  to  the 
pressure  of  the  gas  above  the  liquid  surface.     The  statement  of  this  fact  is 
known  as  Henry's  law.     It  does  not  hold  in  the  cases  of  very  soluble 
gases,  such  as  ammonia.     In  the  following  table  are  given  the  volumes 
of  various  gases  absorbed  under  a  pressure  of  one  atmosphere  by  unit 
volume  of  water  at  different  temperatures: 


t  H  N  O  C02 

0°  .0193  .012035  .04114  1.7967  1050 

10°             "  0.1607  .03250  1.1847  813 

20°             "  0.1403  .02838  0.9014  586 

As  we  may  infer  from  the  above  figures,  dissolved  gases  may  be  driven 
off  by  heat. 

73.  Occlusion.  —  Some  metals  absorb  some  gases  in  large  quanti- 
ties. This  phenomenon  is  sometimes  called  occlusion.  Platinum  will 
absorb  hydrogen  so  violently  that  if  the  platinum  is  already  warm  it  may 
be  raised  to  incandescence;  a  spiral  of  platinum  wire,  for  example,  if 
heated  in  a  Bunsen  flame  may  be  allowed  to  cool  below  red  heat  when  the 
flame  is  extinguished,  and  if  the  gas  be  again  turned  on  the  platinum  will 


42  PROPERTIES    OF    MATTER. 

absorb  hydrogen  from  the  gas  and  become  so  hot  as  to  ignite  the  latter. 
Palladium  may  absorb  more  than  a  thousand  times  its  own  volume  of 
hydrogen,  which  must  in  consequence  be  reduced  to  a  density  comparable 
with  that  of  water.  The  attractive  force  required  to  condense  hydrogen 
to  this  extent  must  be  enormous.  It  is  possible  that  the  hydrogen,  which 
has  many  metalic  properties,  may  form  something  like  an  alloy  with  these 
metals.  Iron  will  absorb  carbon  monoxide  in  considerable  quantities. 

The  gases  absorbed  by  both  liquids  and  solids  may  be  driven  off  by 
heating  or  by  placing  them  in  a  vacuum.  On  heating  water,  for  example, 
air  bubbles  are  seen  to  form  and  rise  throughout  its  entire  mass;  the  same 
effect  will  be  observed  under  the  receiver  of  an  air  pump.  Platinum, 
palladium,  and  iron  will  give  up  their  absorbed  gases  at  high  temperatures, 
or  will  transmit  them  if  they  are  present  on  one  side  only.  If  a  stream  of 
impure  hydrogen  be  passed  through  a  platinum  or  palladium  tube  at  a  high 
temperature,  pure  hydrogen  will  pass  through  the  walls  of  the  tube.  In 
this  way  the  purest  hydrogen  may  be  secured.  Carbon  monoxide  will 
readily  pass  through  hot  iron,  and  may  thus  contaminate  the  atmosphere 
of  a  room  by  passing  through  the  walls  of  a  stove  in  which  the  combus- 
tion is  imperfect,  resulting  in  the  formation  of  this  gas. 

74:.  Adsorption. — Glass  is  apparently  impervious  to  any  known  gas 
or  liquid;  yet  it  is  easily  shown  that  when  a  piece  of  glass  is  heated,  con- 
siderable quantities  of  air,  water  vapor,  and  carbon  dioxide  are  given  off. 
These  gases  seem  to  be  strongly  condensed  on  the  surface  of  the  glass; 
they  may  be  said  to  ' '  wet ' '  it,  and  the  molecules  cling  together  in  much 
the  same  way  that  liquid  water  clings  to  the  glass  that  it  wet's.  This  phe- 
nomenon, which  is  called  adsorption,  shows  itself  in  filling  a  barometer 
tube;  the  tube  must  be  repeatedly  heated  after  being  filled  with  mercury, 
partly  to  drive  off  the  air  absorbed  by  the  mercury,  partly  to  remove  the 
condensed  gases  and  vapors  on  the  glass.  In  the  vacuum  tubes  designed 
to  show  the  electric  discharge  through  gases  at  low  pressures  it  is  almost 
impossible  to  get  rid  of  the  water  vapor  adhering  to  the  \valls  of  the  tube. 
Prolonged  heating  in  the  presence  of  metallic  sodium  or  potassium  seems 
to  be  the  only  effective  method  of  removing  it.  For  the  same  reason 
glass  insulating  stands  for  electrostatic  experiments  usually  fail  to  insulate 
until  they  are  well  dried  by  warming.  It  is  difficult  to  test  Boyle' s  law  at 
very  low  pressures  on  account  of  the  tendency  of  the  gas  to  condense  in 
the  containing  vessel  or  for  already  condensed  gas  to  pass  off.  This  con- 
densation is  clearly  a  case  of  molecular  attraction,  for  different  solids  differ 
very  widely  in  the  extent  to  which  they  condense  gases  on  their  surface. 

Finely  divided  platinum  sponge  will  absorb  much  more  hydrogen  than 
the  same  quantity  of  solid  platinum.  The  amount  of  available  surface 
seems  important,  so  that  in  such  cases  both  absorption  and  adsorption 
seem  to  be  active.  The  same  is  true  of  other  solids,  such  as  charcoal, 
which  absorbs  several  hundred  times  its  own  volume  of  ammonia  gas  and 
large  quantities  of  all  other  gases;  hence  its  value  as  a  deodorizer.  The 
more  porous  or  finely  divided  the  charcoal  is — that  is,  the  more  surface  it 
exposes — the  more  effective  it  is  in  this  respect. 

If  a  letter  or  design  be  firmly  traced  on  a  glass  plate  with  a  soft  stick, 
and  if  the  glass  be  afterward  breathed  upon,  the  water  vapor  in  the  breath 
will  condense  more  freely  on  the  untouched  portions,  bringing  out  the 
design  plainly.  The  scraping  away  of  the  condensed  gases  on  the  glass 
seems  to  affect  its  power  of  condensing  water  vapor.  If  a  coin  be  laid  for 
a  time  on  a  clean  glass  or  metallic  surface,  and  then  removed,  its  image 


VISCOSITY.  43 

may  be  brought  out  in  the  same  way.     These  images  are  called  Moser*  s 
breath  figures. 

75.  Viscosity. — There  is  more  or  less  molecular  friction  in  gases, 
or  between  gases  and  solids,  which  is  called  viscosity  when  it  concerns  the 
friction  between  parts  of  the  same  substance.     The  viscosity  of  gases  is 
strikingly  shown  in  the  transpiration  of  the  gas  through  a  fine  capillary 
tube,  on  the  walls  of  which  the  gas  becomes  adsorbed.    Maxwell  compared 
the  viscosities  of  different  gases  by  observing  the  vibrations  of  a  thin  disc, 
suspended  as  a  torsion  pendulum,   over  a  similar  disc  placed  under  it. 
The  viscosity  of  the  gas  between  the  two  discs  acted  as  a  friction  brake 
to  bring  the  pendulum  to  rest ;    the  less  viscous  the  gas,  the  longer  the 
vibrations  persisted.      Hydrogen  is  the  least  viscous  of  gases;  air  has  about 
twice  as  great  a  viscosity.     It  is  to  this  property  that  the  suspension  in  the 
air  of  fine  dust  and  smoke  particles,  etc.,  is  due.     The  weight  of  these 
particles  diminishes  as  their  volume,  or  as  the  cube  of  their  linear  dimen- 
sions, while  their  surfaces  diminish  as  the  square  of  their  linear  dimensions, 
or  less  rapidly.     When  the  particles  are  very  fine,  therefore,  friction  against 
the  air  becomes  large  as  compared  with  their  weight,  and  they  fall  very 
slowly.     The  friction  increases  very  rapidly  with  velocity;  so  that  meteors 
in  the  upper  atmosphere,  traveling  thirty  or  forty  miles  a  second,  may 
become  incandescent,  even  in  greatly  rarefied  air. 

The  resistance  to  the  motion  of  a  projectile  through  the  air  is  due  to 
several  causes;  partly  to  the  friction,  partly  to  the  mass  of  air  dragged 
along  with  the  projectile,  thus  increasing  its  inertia,  and  partly  to  the 
elastic  resistance  of  the  compressed  air  before  it.  On  account  of  the  dif- 
ferences in  the  density  of  the  air,  and  consequent  differences  in  its  refrac- 
tive power,  instantaneous  photographs  of  projectiles  may  be  made,  which 
show  the  compression  wave  in  front  and  the  eddies  and  wake  behind,  their 
general  appearance  being  strikingly  similar  to  the  water  waves,  eddies,  and 
wake  around  a  rapidly  moving  ferry  boat.  Friction  plays  a  considerable 
part  in  such  phenomena,  causing  a  mass  of  the  fluid  to  be  dragged  along 
with  the  moving  body.(^ 

The  viscosity  of  liquids  is  diminished  with  rising  temperatures ;  mo- 
lasses becomes  very  thin  when  it  is  heated.  With  gases  the  contrary  is 
true ;  the  viscosity  increases  with  the  temperature.  The  reason  for  this 
we  shall  find  suggested  in  the  next  section. 

KINETIC  THEORY  OF  MATTER. 

76.  It  was  once  supposed  that  the  pressure  of  gases  is  due  to  a 
repulsion  between  their  molecules.     If  such  were  the  case,  or  even  if  it  were 
due  to  an  elastic  reaction  against  actual   molecular  contact  (which  sup- 
position seems  to  be  excluded  by  the  unlimited  expansiveness  of  gases),  a 
compressed  gas  would  possess  a  large  amount  of  potential  energy  which 
would  become  kinetic  if  the  gas  were  allowed  to  expand  without  doing 
external  work.     Molecular  kinetic  energy,  however,  is  heat;  consequently, 
a  gas  expanding  into  free  space  should  be  heated.     As  shown  by  the 
experiments  of  Thomson  and  Joule,  which  will  be  discussed  later,  this  is  not 
the  case ;  the  temperature  changes  of  a  gas  expanding  into  a  vacuum  are 
vanishingly  small. 

Hooke,  in  1678,  Daniel  Bernouilli,  in  1738,  and  others  at  various 
times,  suggested  that  the  pressure  of  a  gas  might  be  due  to  the  impact  of 
its  molecules.  Clausius  was  the  first  to  state  the  theory  in  precise  terms. 
(1857). 

(1)   See  Boys,  Nature,  1892. 


44  PROPERTIES    OF    MATTER. 

According  to  this  theory  all  molecules  are  supposed  to  be  in  constant 
motion.  In  gases  and  liquids  they  are  free  to  travel  in  any  direction,  im- 
peded only  by  frictional  forces  ;  in  solids  they  have  only  a  small  degree  of 
vibratory  freedom  about  a  mean  position.  The  diffusion  phenomena  ob- 
served in  fluids  are  strong  evidences  in  favor  of  this  view  ;  and  if  the 
theory  that  heat  is  molecular  kinetic  energy  is  correct,  the  molecules  of 
all  kinds  of  matter  above  a  temperature  of  absolute  zero  must  be  in  motion 
of  some  sort.  Thus  direct  observation  and  agreement  with  another  well- 
supported  theory  both  strengthen  the  validity  of  the  kinetic  theory  of 
matter. 

In  the  case  of  gases  there  are  probably  frequent  impacts  between  the 
molecules  as  they  move  at  random,  the  number  diminishing  as  the 
density  diminishes.  We  have  no  reason  to  believe  that  all  the  molecules 
of  a  given  gas  move  with  the  same  velocity  ;  it  seems,  on  the  contrary, 
exceedingly  probable  that  they  are  moving  in  all  directions  and  with  all 
speeds  between  zero  and  a  maximum  depending  on  the  temperature. 
Furthermore,  these  speeds  are  probably  constantly  subjected  to  sudden 
changes  in  amount  and  direction  of  motion  by  impacts  against  each  other 
and  against  the  walls  of  the  containing  vessel.  Let  us  consider  whether 
the  pressure  exerted  on  the  walls  can  be  explained  as  the  effect  of  motion. 

In  a  given  gas  all  the  molecules  have  the  same  mass  m.  If  these 
molecules  behave  like  elastic  spheres,  and  if  one  having  a  component 
velocity  u^  in  the  X  direction  strikes  the  wall  of  a  cubical  enclosure  viith 
sides  of  length  /,  it  will  rebound  with  the  same  velocity  ut.  The  impul- 
sive pressure  on  the  wall  during  the  time  of  contact  /  is/,,  and  by  the 
principle  of  change  of  momentum  we  know  that 


(75)  .  m  \UT  -(-«,)]  =  2muT  =/,  L 

If  the  molecule  meets  no  other  molecule,  but  rebounds  freely  back  and 
forth  between  opposite  walls,  collisions  with  the  wall  will  occur  at  intervals 
equal  to  2/  /  uy  and  the  number  of  impacts  per  second  will  be  n  =  ut  /  2/. 
A  rapid  succession  of  impacts  is  equivalent  in  its  effects  to  a  steady 
pressure,  if  exerted  on  bodies  having  inertia,  such  as  the  walls  of  a  vessel. 
We  might,  for  example,  imagine  a  ballistic  pendulum  permanently  and 
steadily  deflected  by  the  rain  of  bullets  from  a  rapid-fire  gun.  The 
equivalent  pressure  is  the  time  average  of  the  impulsive  forces,  or 


and  if  T  is  taken  as  one  second, 

mu. 


(77) 


I 

The  partial  pressures  given  by  the  molecules  moving  with  other  velocities 
can  be  expressed  in  the  same  way,  and  the  total  pressure  p  is 

m 

There  is  an  ideal  velocity  u,  called  the  velocity  of  mean  square,  which 
is  defined  by  the  relation, 

(79*)         ,  #V  =  */  +  V  +  *3a  •  •  •  ».'• 

Perhaps  no  molecule  may  have  a  velocity  of  this  exact  value,  but  if  all  the 


KINETIC    THEORY    OF    MATTER.  45 

molecules  did  have  this  velocity  the  total   pressure  effect  would  be  the 
same  as  in  the  actual  case. 
This  relation  now  gives 

(80)  ^mu*=p. 

Since  we  are  dealing  with  a  column  of  unit  cross  section  and  of  length 

N 
I,  N  j  I  is  the  total  number  of  molecules  per  cubic  cm.,  and-y-  m  =  p,  the 

density  of  the  gas,  hence 

(81)  -^mu*  =  PU2=p. 

Furthermore,  the  law  of  averages  applied  to  large  numbers  would  lead 
us  to  infer  that  the  velocities  of  mean  square  v  and  w  in  the  Kand  Z 
directions,  are  each  equal  to  u,  and  that  the  relation  of  these  components 
to  the  actual  speed  in  the  line  of  motion  must  be 

(82)  wa  +  z/a  +  o/a  =  3«a  =  F2, 
from  which 


Multiplying  by  v,  the  volume  of  the  gas, 

(83)  pv  =  i  MV*  =  constant. 

This  evidently  corresponds  to  Boyle's  law. 

The   mechanical   theory   of   heat   suggests    the   probability   that   the 

temperature  of  a  mass  of  gas  or  other  substance  is  proportional  to  the 

mean  kinetic  energy  of  its  molecules.      F2  would  then  be  proportional  to 

the  absolute  temperature,  and  the  product  pv,  being  also  proportional  to 

F2,  would  likewise  be  proportional  to  the  absolute  temperature,  or 

(84)  PV  =  ^MV*  =  MRT. 

The  kinetic  theory  is  thus  a  satisfactory  explanation  of  Charles'   law. 

If  the  average  molecular  kinetic  energy  of  two  different  gases  at  the 
same  temperature  were  not  the  same,  we  should  expect  their  temperature 
to  change  if  they  were  mixed,  as  a  result  of  the  equalizing  effect  of 
impacts.  Such  a  change  is  not  observed,  hence 

(85)  \mT  V^  =  \m2  F2*. 

If  we  consider  the  same  two  gases  separately  at  the  same  pressure 

(86)  P  =  \N,  m^  V*  =  $N2m2  V;. 
Comparing  the  last  two  equations, 

(87)  N^  =  N^ 

or  the  number  of  molecules  per  unit  volume  of  all  gases  at  the  same 
pressure  and  temperature  is  the  same.  This  is  Avogadro'  s  law,  first  deduced 
by  an  entirely  different  method  from  chemical  considerations. 

A  gas  when  rapidly  compressed  becomes  heated,  because  work  is 
expended  against  the  impacts  of  its  molecules,  increasing  their  kinetic 


46  PROPERTIES    OF    MATTER. 

energy;  when  they  do  work  against  pressure  in  expanding  they  lose 
kinetic  energy  and  are  cooled.  If  they  expand  against  a  vacuum,  and  if 
there  are  no  intermolecular  forces,  no  work  is  done  and  there  will  be  no 
change  of  temperature.  It  will  be  found  that  the  gases  investigated  by 
Thomson  and  Joule  only  approximately  fulfilled  these  conditions,  indicat- 
ing that  there  are  slight  molecular  forces,  and  that  there  is  n.o'  perfect  gas, 
or  one  which  perfectly  fulfills  the  gaseous  laws. 

Dalton's  law  is  evidently  in  accord  with  the  kinetic  theory. 

We  may  deduce  the  velocity  of  mean  square  for  different  gases  from 
the  relation 

(88) 

From  this  we  may  calculate  the  velocity  of  mean  square  for  hydrogen  at 
0°  C.,  and  find  it  to  be  about  1,860  meters  per  second;  that  of  oxygen  is 
465.  The  mean-square  velocity  of  different  gases  varies  inversely  as  the 
square  root  of  the  density,  as  in  the  cases  of  efflux  and  diffusion,  but  in 
each  case  the  numerical  value  of  the  velocity  is  different.  In  the  case  of 
efflux  there  is  a  single  definite  mass  velocity;  in  the  other  case  the 
molecular  velocities  may  range  from  zero  to  high  values;  but  we  can  at 
least  say  that  on  the  average  the  velocity  is  inversely  proportional  to  the 
square  root  of  the  density  of  the  gas. 

Many  phenomena  connected  with  liquids  and  solids  make  it  seem 
probable  that  in  them  likewise  there  is  constant  molecular  motion,  but  on 
account  of  the  proximity  of  the  molecules  the  molecular  forces  and 
constraints  involved  make  the  problem  too  difficult  to  be  treated  mathe- 
matically. 

It  is  probable  that  the  kinetic  theory  of  gases  can  never  be  directly 
verified.  The  fundamental  assumption  that  molecules  are  spherical  and 
perfectly  elastic  bodies  is  almost  certainly  false.  The  hypothetical  velocity 
of  mean  square  is  one  which  perhaps  not  a  single  molecule  has.  Neverthe- 
less, the  kinetic  theory  fulfills  all  the  purposes  of  a  useful  theory.  It 
accounts  for  gaseous  pressure;  for  diffusion  of  gases  and  liquids;  for  the 
evaporation  of  liquids,  which  is  the  result  of  isolated  molecules  breaking 
through  the  surface  on  account  of  their  momentum ;  and  for  the  fact  that 
all  these  effects  are  increased  by  rise  of  temperature.  It  also  throws  light  on 
the  fact  that  the  viscosity  of  gases  increases  with  rise  of  temperature,  while 
that  of  liquids  diminishes.  In  gases  the  apparent  frictional  effect  cannot 
be  the  result  of  actual  contacts.  We  must  assume  that  the  molecules  in 
one  stratum  are  constantly  passing  into  adjacent  strata  and  conversely.  If 
there  is  relative  motion  of  the  strata,  the  result  will  be  that  the  molecules 
passing  from  a  stratum  at  rest  into  one  in  motion  will  slightly  retard  the 
motion  of  the  latter,  as  the  moving  molecules  must  impart  some  momentum 
to  those  relatively  at  rest;  those  passing  from  the  stratum  which  is  in  motion 
will  in  a  similar  manner  set  the  other  stratum  in  motion.  The  consequence 
is  a  dragging  effect,  just  as  in  the  imaginary  case  of  two  boats  moving  with 
different  velocities  parallel  to  each  other.  If  heavy  objects  be  constantly 
thrown  backward  and  forward  between  them  the  exchange  of/  momentum 
would  finally  equalize  their  velocities.  In  liquids,  on  the  contrary,  the 
friction  is  due  to  actual  contacts  and  attractions;  the  increased  velocities 
due  to  rise  in  temperature  tend  to  break  down  molecular  constraints  and 
increase  freedom  of  motion. 

All  these  facts  give  a  high  degree  of  probability  to  the  kinetic  theory, 


HYDROSTATICS HYDRODYNAMICS.  47 

although  its  exact  details  can  be  understood  but  vaguely.  We  must  never 
lose  sight  of  the  fact,  however,  that  it  is  only  a  theory,  and  that  it  may  in 
time  be  replaced  by  a  more  satisfactory  one;  that  is  to  say,  one  which  will 
closely  associate  and  explain  a  greater  number  of  phenomena  than  those 
which  have  led  us  to  the  kinetic  theory  as  a  comprehensive  description. 

SPECIAL  PROPERTIES  OF  LIQUIDS. 

77.  Those  fluids  which  have  a  nearly  constant  volume,   very  slightly 
affected  by  pressure  or  heat,  and  which  may  in  consequence  have  a  free 
surface,  are  called  liquids. 

The  mechanics  of  liquids  may  be  divided  into  two  branches, 
Hydrostatics,  which  deals  with  liquids  at  rest;  Hydrodynamics,  which 
deals  with  liquids  in  motion. 

78.  Hydrostatics. — The  fundamental  laws  of  hydrostatics  are  those 
of  Pascal  and  Archimedes,  which  apply  to  all  fluids,  whether  gaseous  or 
liquid.     These  have  already  been  discussed. 

Equilibrium. — An  immersed  body  is  in  equilibrium  when  the  center  of 
gravity  of  the  body  lies  vertically  below  the  center  of  buoyancy  (the 
center  of  gravity  of  the  displaced  liquid).  Equilibrium  may  also  be 
stable  in  some  cases  when  the  center  of  gravity  is  vertically  above 
the  center  of  buoyancy,  as  in  a  floating  board.  If  on  slightly  displacing 
the  body  so  that  a  vertical  line  through  the  center  of  buoyancy  cuts  the 
vertical  through  the  original  center  of  buoyancy  and  the  center  of 
gravity,  the  point  of  intersection  lies  above  the  center  of  gravity,  equi- 
librium is  stable;  if  they  coincide,  neutral;  if  the  point  of  intersection  is 
below  the  center  of  gravity,  unstable.  The  point  of  intersection  is  called 
the  metacenter,  and  in  ships  it  is  important  that  it  should  be  as  high  as 
possible,  to  secure  stability. 

The  surface  of  the  liquid  at  rest  must  always  be  normal  to  the  result- 
ant of  the  forces  acting  on  it,  or  level  when  gravity  alone  acts.  Inequali- 
ties of  atmospheric  pressure  may  produce  perceptible  differences  of  level 
between  different  parts  of  seas  and  lakes.  High  water  or  low  water  may 
be  thus  maintained  by  winds.  If  a  cylindrical  vessel  containing  a  liquid  be 
rotated  at  a  uniform  rate  about  its  vertical  axis,  the  surface  of  the  liquid 
will  assume  the  shape  of  a  paraboloid  of  revolution,  owing  to  centrifugal 
action. 

79.  Hydrodynamics. — Since  a  liquid  is  practically  incompressible, 
the  quantity  flowing  through  every  cross-section  of  a  stream  must  be  the 
same,  and  the  speed  of  a  liquid  at  a  given  point  must  be  inversely  as  the 
cross-section  of  the  stream  at  that  point. 

If  a  liquid  flows  from  an  aperture,  the  pressure  producing  flow,  if 
gravity  alone  acts,  is  p  =  pgh,  h  being  the  height  of  liquid  above  the  orifice, 
or  the  "head." 

80.  Torriceltf  s   Law. — As  shown  in  discussing  efflux  of  gases,   if 
the  work  done  in  driving  a  mass  of  fluid  through  an  orifice  is  entirely 
converted  into  kinetic  energy, 

(89)  s*  =  2pjt>  =  Zgh. 

As  this  expression  does  not  involve  density,  it  follows  that  all  liquids, 
from  ether  to  mercury,  flow  with  the  same  speed  under  the  same  head. 
This  is  not  the  case  with  gases. 


48  PROPERTIES    OF    MATTER. 

81.  Jets. — If  we  neglect  the  effect  of  friction  and  the  resistance  of 
the  air,  a  jet  of  liquid,  like  a  projectile,  will  fall  in  a  parabolic  path. 

Owing  to  lateral  flow,  a  jet  does  not  have  a  cylindrical  form,  but  con- 
tracts to  a  minimum  cross-section  just  above  the  orifice.  This  is  called 
the  vena  contracta.  The  volume  of  liquid  flowing  out  is  equal  to  the 
area  of  vena  contracta  multiplied  by  the  speed.  Experiment  shows  that 
for  an  orifice  of  area  a  in  a  thin  wall  V=  .62as  per  second.  The  volume 
of  flow  is  altered  by  mouth-pieces,  or  ajutages.  If  these  project  inward, 
flow  is  diminished;  if  outward,  it  may  be  increased.  If  cylindrical  and  wet 
by  the  liquid,  there  is  a  suction,  which  tends  to  enlarge  the  vena  contracta, 
causing  a  flow  of  about  .  Sas.  With  a  conical  ajutage  the  flow  may  be  .  9as. 

82.  Flow   in   Pipes. — If  a  liquid  flows  in  a  horizontal  pipe  under  a 
given  head,  the  velocity  in  the  pipe  will  not  correspond  to  that  indicated 
by  the  head,  owing  to  the  fact  that  some  energy  is  expanded  in  overcom- 
ing friction.     The  pressure  indicated  by  vertical  manometers  called  piezo- 
meters,   communicating   with   the   pipe,    gradually  diminishes  in   going 
toward  the  end  of  the  pipe,   vanishing  at  the  point  of  exit.     The  actual 
speed  of  the  fluid  is  that  due  to  a  certain  head  h  less  than   H.     The  pres- 
sure at  any  point  may  be  ascribed  to  an  imaginary  head  h'.     At  the  point 
where  the  liquid  enters  the  pipe  h  +  h'  =  H.     The  pressure  head  at  any 
point  is  modified  by  the  resistance  of  the  entire  pipe  beyond  that  point. 
If  the  pipe  is  of  variable  cross-section,  the  velocity  at  any  point  is  inversely 
as  the  cross-section,  and  piezometers  show  that  the  pressure  head  is  great- 
est where  the  velocity  head  is  least — i.  e. ,  as  shown  in  the  case  of  gases, 
the  pressure  is  greatest  when  the   velocity  is  least.     As  expressed   by 
Bernouilli's  law. 

/  +  \  P  s2  =  constant, 

or  as  it  may  also  be  stated,  the  sum  of  the  potential  and  the  kinetic 
energies  of  a  unit  volume  remains  constant. 

The  speed  of  a  liquid  is  impeded  by  viscosity.  A  stream  flows  most 
slowly  near  the  sides  and  bottom  of  its  channel. 

Flow  through  capillary  tubes  is  greatly  affected  by  viscosity.  Poiseuille 
found  that  the  rate  of  flow  is  proportional  directly  to  the  fourth  power  of 
the  radius  and  inversely  to  the  length  of  the  tube. 

83.  Practical  Application. — A    moving    stream    by   reason    of  its 
momentum  can  if  checked  produce  an  instantaneous  pressure  greatly  in 
excess  of  the  actual  hydrostatic  pressure.     This  is  utilized  in  the  case  of 
the  water  ram,  by  which  water  may  be  raised  higher  than  its  source  by 
the  momentum  of  a  larger  mass  of  water.     The  pressure  of  the  stream 
R  closes  a  valve  v,  and  the  momentum  opens  w  and  compresses  air  in  L. 
This  closes  w;  v  again  opens  when  the  flow  ceases  while  the  compressed 
air  raises  the  water  above  N.     The  kinetic  energy  of  moving  water  is 
transformed  into  work  by  means  of  various  forms  of  water  wheels,  such 
as  the  overshot,   undershot,  and  turbine  (illustrated  by  Barker's  mill). 
The  more  nearly  the  water  loses  its  entire  velocity,  the  more  efficient  the 
wheel. 

84.  Vortices. — In   flowing    over   obstructions    or    around    corners, 
water   acquires  a  rotational    motion,    producing    vortices    or    whirlpools. 
Similar  phenomena  in  gases  are  illustrated  by  smoke  rings,  and  also,  on  a 
larger   scale,    by    cyclones.       It    is    by    friction    that   such   vortices    are 
produced  and  brought  to  rest.     Helmholtz  has  shown  that  in  a  "perfect" 


COMPRESSIBILITY.  49 

fluid — that  is  to  say,  one  having  no  viscosity — it  would  be  impossible  to 
originate  vortices  by  any  means  known  to  us,  or,  assuming  such  vortices 
to  exist,  it  would  be  impossible  for  us  to  destroy  them.  These  properties 
suggested  to  Lord  Kelvin  the  vortex-atom  theory  of  matter. 

MOLECULAR   FORCES   IN   LIQUIDS. 

85.  Compressibility. — Liquids  are  all  slightly  compressible,  by  an 
amount  which  is  difficult  to  measure.  Lord  Bacon  tried  to  determine  the 
compressibility  of  water  by  compressing  it  in  a  closed  hollow  leaden 
sphere;  the  Florentine  academicians  (1692)  repeated  the  experiment  with 
a  gilded  silver  sphere;  in  each  case  the  water  escaped  through  invisible 
pores  of  the  metal.  Canton,  about  1762,  was  able  to  prove  that  water  is 
compressible.  Oersted,  about  1822,  devised  an  improved  form  of 
piezometer,  or  pressure  measurer,  which  gave  more  reliable  results.  The 
thermometer-shaped  vessel  V  containing  the  water  was  immersed  in  a 
larger  vessel  also  containing  water,  to  which  the  pressure  was  applied. 
The  glass  walls  of  Fwere  thus  subjected  to  equal  pressures  without  and 
within,  while  the  change  of  volume  of  its  contents  could  be  determined 
by  the  motion  of  a  pellet  of  mercury  in  the  capillary  neck.  If  the 
volume  of  the  bulb  is  V,  that  of  each  division  of  the  stem  v,  and  the 
water  is  compressed  from  the  mih  to  the  nth  division  by  a  pressure  of  P 
atmospheres,  the  reduction  of  unit  volume  by  one  atmosphere  pressure, 
or  the  compressibility,  is 

(m-n)v 
~  P(V+mvy 

The  volume  of  V^  is,  however,  on  the  whole  diminished  by  the  pressure 
which  thins  its  walls  (as  may  readily  be  seen  by  imagining  the  vessel  a 
part  of  a  solid  block  of  glass  submitted  to  uniform  external  pressure). 
This  diminishes  the  apparent  contraction  of  the  water,  so  the  contraction 
c'  of  the  vessel  per  unit  volume  for  each  atmosphere  must  be  added  to  c. 
Where  c'  —  c  there  would  be  no  apparent  change  of  volume  in  the  water. 
The  compressibility  of  most  liquids  increases  with  the  temperature. 
That  of  water,  however,  has  a  minimum  about  61°.  Below  are  some 
values  of  c. 
Water—  Ether— 

0°  0000503  0°  00011      Mercury 0000029 

16°  450  14°  16 

61°  389          100°  56      Liquid  CO2 00590 

77°.4 398 

99°.2 409 

Salts  dissolved  in  water  usually  diminish  its  compressibility,  as  shown 
below  for  sodium  chloride  solution : 

Per  Cent.  NaCl  c 

5  .0000455 

10  397 

20  306 

25  258 

Water  is  about  25  times  as  compressible  as  copper,  40  times  more  than 
iron,  80  times  more  than  nickel.  Compressibility  diminishes  as  pressure 
increases,  indicating  that  there  is  a  definite  limit  to  the  process. 

Liquids,  from  which  all  air  has  been  removed  by  boiling,  can  support 
a  tension,  stretching  without  rupture.  (Berthelot).  Pure  water  will  resist 


50  PROPERTIES     OF     MATTER. 

a  tension  of  50  atmospheres  weight  ;  sugar  solution,   nearly  100  atmos- 
pheres.    The  sugar  seems  to  increase  cohesiveness. 

86.  Cohesion  and  Adhesion  are  the  names  applied  to  the  forces 
which  bind  together   the   molecules    of  liquids    and   solids.       They  are 
sensible  only   through   very   small   distances.      The  adhesive   properties 
of  glue,   gilding,   silvering,   etc.  ,   depend    on    intimate  contact.       Similar 
forces  exist  between  liquids  and  solids  and  within  liquids.      Clean  glass 
is  wet  .by  water  ;  that  is,   the  water  clings  to  it  with  such  force  that  it 
cannot  be  directly  removed,  this  adhesion  being  of  course  much  greater 
than  the  cohesion  of  water.       In  mercury  the  cohesion  is  greater,  but  it 
still  requires  considerable  force  to  detach  a  glass  plate  from  mercury, 
which  does  not  wet  it  ;  much  more  force  to  detach  a  copper  plate,  which 
the  mercury  does  wet.     (Amalgamation). 

87.  Yiscosity.  —  Although  all  liquids  are  mobile,  the  rates  of  flow  are 
very  different,   owing  to  internal  forces,  frictional  or  otherwise.      If  we 
imagine  two  parallel  surfaces  in  a  liquid  moving  with  relative  velocity  v, 
the  viscous  resistance  R  offered  to  the  relative  motion  will  be  found  to  be- 
proportional  to  the  velocity  v,  the  area  of  the  surfaces  s,  and  the  coefficient 
of  vicosity  /,  and  inversely  proportional  to  the  distance  d,  between  the 


(9.)  *or/ 

The  viscous  resistance  offered  by  parallel  layers  of  liquid  to  relative 
motion  may  be  compared  with  the  resistance  offered  by  a  book  to  a  tan- 
gential force  tending  to  slide  its  leaves  over  each  other. 

For  glycerine,  /=  .  00238;  for  olive  oil,  .001  ;  for  water,  .0000137. 
Salt  solutions  are  usually  more  viscous  than  water. 

A  large  part  of  the  resistance  to  the  motion  of  a  ship  is  due  to  the 
viscous  resistance  between  the  stationary  water  and  the  water  film  on  her 
hull.  The  efficiency  of  lubricants  depends  on  the  existence  of  films  of  the 
lubricant  between  and  wetting  the  moving  solid  parts  of  machinery  ; 
the  viscosity  between  the  films  of  oil  is  less  than  the  friction  between  the 
dry  solids.  Viscosity  brings  winds  and  waves  to  rest.  Rise  of  tempera- 
ture decreases  viscosity,  that  is,  increases  molecular  mobility. 

Small  particles  will  remain  suspended  in  liquids  for  some  time,  owing 
to  viscous  resistance  to  fall.  Enormous  quantities  of  soil  are  thus  trans- 
ported into  the  ocean.  The  Mississippi  deposits  in  its  delta  every  year 
a  mass  of  silt  about  one  mile  square  and  240  feet  deep,  besides  a  much 
larger  quantity  carried  by  flotation  or  pushing. 

88.  Surface  Tension.  —  The  unbalanced  attractions  acting  on  the 
molecules  on  the  surface  of  a  mass  of  liquid  pull  them  toward  the  center 
and  toward  each  other.  The  effect  is  as  though  the  liquid  were  contained 
in  a  skin  or  elastic  membrane,  which  takes  the  shape  having  the  smallest 
surface  —  a  sphere.  This  is  illustrated  in  water  drops,  mercury  drops, 
melted  glass  and  sealing-wax,  and  the  manufacture  of  shot.  The  perfect 
sphericity  of  raindrops  is  proved  by  the  circular  shape  of  rainbows.  The 
action  of  gravity  ordinarily  masks  this  result,  but  Plateau  overcame  this 
difficulty  by  suspending  drops  of  oil  in  a  mixture  of  water  and  alcohol 
having  the  same  density  as  the  oil,  which  then  assumed  a  spherical  shape. 

The  existence  of  surface  films  may  be  shown  by  sprinkling  lycopodium 
powder  over  clean  water.  On  dipping  a  slightly  greasy  glass  rod  in  the 


SURFACE    TENSION.  51 

water,  the  film  is  weakened  and  retreats  with  the  powder,  leaving  a  clear 
space  around  the  rod.  Heat  will  cause  the  same  result.  Alcohol  and 
ether  likewise  weaken  the  tension,  causing  the  water  to  retreat,  like  a 
stretched  sheet  of  rubber  weakened  at  a  given  point.  Pieces  of  camphor 
will  dart  along  the  surface  of  pure  water  in  all  directions,  owing  to  local 
variations  of  tension  due  to  the  dissolved  camphor. 

Pure  water  has  a  stronger  surface  tension  than  contaminated  water, 
but  separate  films  and  bubbles  cannot  be  formed  from  it.  A  vertical  film 
must  necessarily  have  a  varying  tension  at  different  points,  while  that  of 
water  is  constant;  consequently  equilibrium  is  impossible.  In  order  to 
form  films  and  bubbles  heterogeneous  mixtures,  such  as  solutions  of  soap 
and  glycerine  in  water,  must  be  used.  In  these  the  tension  can  adjust 
itself  to  the  necessary  conditions.  Such  a  solution  has  a  great  superficial 
tenacity  as  distinguished  from  tension,  and  with  it  soap  bubbles  may  be 
blown  and  films  formed  on  frames  of  various  shapes,  the  films  always 
taking  the  minimum  possible  surface  under  the  conditions.  If  a  loop  of 
silk  is  placed  on  such  a  film,  which  is  then  broken  inside  the  loop,  the 
latter  will  at  once  become  an  exact  circle.  These  films  will  sustain  con- 
siderable weight. 

Surface  tension  is  defined  as  the  force  acting  across  unit  length  in  a 
surface,  or  by  one-half  this  quantity  in  a  film,  since  it  has  two  surfaces. 
The  contraction  of  liquid  films  is  another  example  of  potential  energy 
tending. to  a  minimum. 

The  equilibrium  position  of  three  surfaces  in  contact  is  determined  by 
the  relative  surface  tensions  between  each  pair  of  fluids.  Consider  a 
drop  of  liquid  on  the  surface  of  another  liquid  in  air.  Three  forces, 
x  T2,  2  Ty  3  TIt  act  on  any  element  on  the  common  boundary  line.  These 
forces  will  be  in  equilibrium  when  the  angles  between  them  are  such  that 
the  resultant  of  any  pair  is  equal  and  opposite  to  the  other  force.  If  one 
force  exceeds  the  sum  of  the  other  two,  equilibrium  is  impossible. 
Suppose  a  Tw  >  w  T0  H-  0  T&  as  is  the  case  when  a  drop  of  oil  floats  on  water. 
The  drop  of  oil  will  be  drawn  out  in  a  thin  film  covering  the  water. 
Likewise  in  the  case  of  glass,  water,  and  air, 

(92)  a7;>g7;  +  w7;, 

and  the  water  surface  spreads  over  the  glass. 

This  effect  is  also  observed  in  a  vacuum,  but  air  has  probably  some 
small  influence  on  the  result  which  cannot  be  readily  determined,  since 
even  in  a  vacuum  there  is  generally  a  film  of  air  or  other  gas  condensed 
on  the  solid. 

Oil  on  water  prevents  waves,  partly  by  reducing  the  friction  whbh 
enables  wind  to  heap  up  waves,  partly  by  reducing  the  surface  tension 
which  holds  a  heap  together,  as  a  bag  might  hold  sand. 

The  tension  of  a  bubble  compresses  the  air  within  it,  so  that  a  puncture 
is  followed  by  a  collapse.  A  candle  may  be  blown  out  in  this  way. 

89.  Capillarity  deals  with  various  phenomena  due  to  surface 
tension,"  such  as  the  ascent  or  depression  of  liquids  in  small  tubes.  If  the 
solid  is  wet  by  the  liquid,  the  latter  spreads  over  the  surface,  dragging 
with  it  the  surface  film  of  the  liquid  until  the  weight  of  the  column  of 
liquid  balances  the  force  due  to  tension.  In  general,  the  surface  assumes 
a  definite  contact  angle  with  the  solid.  If  6  is  this  angle,  h  the  height  of 


52  PROPERTIES     OF     MATTER. 

the  column,  and  p  the  density  of  the  liquid,  when  equilibrium  is  estab- 
lished— 

2  irrTcos  e  =  irr2hpg. 

Therefore, 

(93)  T  =  r  hpg  I  2  cos  e,  or  h  =  2  Tcos  0  /  rs>g, 

or  the  height  varies  inversely  as  the  radius  of  the  tube  at  the  point  to 
which  the  liquid  rises.  T  may  be  measured  in  this  way.  If  h  is  the 
height  to  which  a  liquid  will  rise  in  a  given  tube,  the  column  which  it  will 
hold  when  freely  suspended  is  h-\-  h^  h^  being  the  height  sustained  by 
pressure  due  to  curvature  of  lower  drop.  If  the  length  of  column  is  /?,  the 
bottom  is  plane;  if  less  than  h,  concave.  Between  two  flat  plates, 
similarly,  the  height  may  be  proved  to  be  inversely  as  the  distance  between 
the  plates.  If  they  are  inclined  to  each  other,  this  leads  to  the  relation 
0ti  =  constant,  or  the  liquid  rises  between  them  in  the  form  of  an  equi- 
lateral hyperbola.  Liquids  which  do  not  wet  the  solid  are  depressed 
according  to  the  same  law  —  e.  g.,  mercury  in  glass. 

The  mechanical  structure  and  size  of  soil  particles  play  an  important 
part  in  soil  physics,  as  the  distribution  of  moisture  by  capillary  action  is 
thereby  determined. 

The  contact  angle  in  the  case  of  water  and  glass  is  zero;  for  mercury 
and  glass,  about  135°.  In  a  globular  vessel  of  glass  there  is  a  certain 
point,  for  which  the  tangent  plane  has  a  slope  of  about  55°  with  the 
vertical,  at  which  the  curvature  of  the  mercury  surface  disappears. 
Above  this  point  it  becomes  concave.  The  contact  angle  may  be  found 
in  any  case  by  dipping  a  plate  of  the  solid  in  the  liquid  and  inclining  it 
until  the  curvature  disappears  on  one  side.  The  inclination  to  the  vertical 
is  the  contact  angle. 

90.  To  find  the  pressure  inside  a  bubble,  consider  the  energy 
change  due  to  a  slight  expansion.  The  work  done  by  the  pressure  must 
be  equal  to  the  gain  in  potential  energy  due  to  increased  surface,  or 


when  the  increase  of  radius  is  infinitesimal.  (2  T  is  used  because  there  are 
two  surfaces). 

The  formula  may  be  deduced  in  another  way.  If  /  is  the  pressure  on 
any  diametral  plane,  the  total  pressure  must  just  balance  the  total  tension 
around  the  circumference,  that  is, 

(94)  rr^p  =  ^r  T,  p  =  2— 

or  twice  this  value  for  a  bubble. 

The  same  expressions  are  used  to  determine  the  tension  required  in 
steam-boilers. 

When  a  soap  film  is  exposed  to  equal  pressures  on  both  sides,  the 
condition,  1  />  +  1  /  r'  =  0  must  be  satisfied. 

In  the  case  of  melted  solids  T  may  be  approximately  determined  by 
melting  the  end  of  a  rod  of  known  radius  and  weighing  the  drop  which 
falls.  The  tension  around  the  circumference  just  before  falling  may  be 
put  equal  to  the  weight  of  the  drop  : 


(95)  2<rrT=  mg 


SURFACE    TENSION.  53 

As  shown  above,  the  resultant  pressure  acts  towards  the  center  of  curva- 
ture of  a  surface  and  is  inversely  proportional  to  the  radius  of  curvature. 
The  liquid  beneath  a  concave  meniscus  is  therefore  in  a  state  of  tension, 
or  "negative  pressure,"  measured  per  unit  surface  by  p  =  2  T/  r.  In 
the  case  of  a  tube  wet  by  a  liquid  the  contact  angle  is  approximately  zero, 
and  the  concavity  becomes  hemispherical,  so  that  r  is  the  radius  of  the 
tube.  Putting  the  hydrostatic  tension  equal  to  the  weight  of  the  column 
sustained,  we  have,  when  equilibrium  is  atttained  : 

(96)  2  Tjr=Pgh 

as  before  shown  by  a  different  method. 

Experiment  shows  that  two  small  floating  objects,  both  wet  or  both 
not  wet  by  a  liquid,  are  apparently  attracted  ;  if  one  is  wet  and  the  other 
not,  there  is  an  apparent  repulsion.  In  such  cases  the  horizontal  com- 
ponents of  the  surface  tensions  are  in  evident  equilibrium  ;  the  resultant 
motion  is  caused  by  the  hydrostatic  pressures  and  tensions  due  to  the 
different  levels. 

A  film  of  water  between  glass  plates  draws  them  together  ;  a  drop  of 
mercury  acts  as  an  elastic  cushion  between  them. 

If  water  be  placed  in  a  U-tube,  one  branch  of  which  is  a  short  capil- 
lary, the  meniscus  in  the  small  tube  will  be  concave,  flat,  or  convex  accord- 
ing to  the  height  of  the  water  in  the  long  branch.  The  hydrostatic  pres- 
sures at  the  same  level  in  both  branches  must  be  the  same,  or 

r 

91.  Below    are    approximate   values   of  T  in  dynes   at   the   inter- 
surfaces  of  some  liquids  at  ordinary  temperatures  : 

Distilled  water-air 77 

Mercury-air 490 

Mercury-water 370 

Petroleum-air 2  '> 

Water-petroleum  oo 

These  values  vary  greatly  with  the  purity  of  the  liquids. 

The  ripples  on  liquids  are  due  to  surface  tension,  the  wavelets  running 
along  the  surface  in  the  same  way  that  waves  are  propagated  along 
stretched  elastic  cords.  The  above  results  were  determined  in  terms  of 
density  of  the  liquid  and  the  wave-length  of  these  disturbances. 

92.  A  cylinder  of  liquid  becomes  unstable  when  its  length  is  greater 
than  its  circumference,  for  it  may  then  have  a  smaller  surface  by  breaking 
up  into  drops.     A  jet  thus  breaks  up  into  drops,  which,  under  the  action 
of  surface  tension,   vibrate  between  the  forms  of  an  oblate  and  a  prolate 
spheroid.     A  jet  ordinarily  appears  to  have  an  undulating  contour.     A 
view  by  the  instantaneous  light  of  the  electric  spark  shows  that  this  shape 
is  due  to  the  formation  of  contractions  preceding  rupture  and  to  the  chain 
of  separate  drops  of  the  shapes  referred  to  which  follow  rupture. 

93.  Effect  of  Temperature. — Heat  diminishes  all  kinds  of  molecu- 
lar cohesiveness,  and  consequently  surface  tension  diminishes  with  rise  of 
temperature.  If  TG  be  the  value  at  0°  , 

T=  T0(\-cf) 
where  c  has  the  value  :  water,  .0019  ;  alcohol,  .0024  ;  ether,  .0047.     For 


54  PROPERTIES     OF     MATTER. 

each  liquid  it  is  evident  that  there  is  a  certain  temperature  at  which  the 
surface  tension  would  vanish.  The  significance  of  this  will  be  shown  in 
discussing  critical  temperatures. 

94.  Surface  tension  of  solutions. —  Impurities  (mechanical  mix- 
tures) in  most  cases  reduce  surface  tension  ;  but  in  definite  solutions  of 
salts,  such  as  that  of  sodium  chloride  in  water,  the  surface  tension  is 
usually  increased  in  proportion  to  the  concentration.  This  is  one  of  a 
number  of  cases  which  indicate  that  salts  in  solutions  tend  to  increase  the 
cohesiveness  of  the  solvent.  They  generally  diminish  the  volume,  or  pull 
the  molecules  together  ;  they  raise  the  boiling  point  ;  they  increase  vis- 
cosity of  liquids  ;  they  increase  surface  tension. 


References. — Tait,  Properties  of  Matter  ;  Boys,  Soap  Bubbles  ;  Mach,  Popu- 
lar Scientific  Lectures — The  Forms  of  Water;  Wm.  Thomson,  Popular  Lectures, 
Vol.  I— Capillary  Attraction. 

QUESTIONS  AND  PROBLEMS. 

48.  Why  are  shot  and  raindrops  not  flattened  by  their  weight  while  falling  ? 

49.  Explain  the  action  of  benzene  or  a  hot  iron  in  removing  grease  spots. 

50.  A  drop  of  water  is  placed  in  a  small  conical  horizontal  glass  tube.     What 
will  it  do  ?    What  will  mercury  do  under  the  same  circumstances  ? 

51.  A  little  water  lies  in  a  uniform  horizontal  glass  tube,  and  one  end  of  the 
column  is  warmed.     What  will  it  do  ? 

52.  Blow  on  warm  soup  or  chocolate  and  observe  effect.     Explain. 

53.  Hold   a    camel's  hair  brush  in  water,  then  take  it  out.     Observe  and 
explain  difference  of  appearance  in  the  two  cases. 

54.  How  far  will  water  rise  and  mercury  fall  in  a  glass  tube  of  .01  cm. 
diameter  ? 

55.  What  is  the  object  of  the  slit  in  a  steel  pen  ? 

56.  Why  will  ink  spread  in  ordinary  printing  paper  and  not  spread  in  writing 
paper  ? 

57.  What  is  (approximately)  the  density  of  sea  water  10,000  feet  below  the 
surface,  at  0°  ? 

58.  Explain  the  washing  process  of  separating  gold  from  gravel ;  the  blast 
process  of  winnowing  grain.     Is  the  same  principle  used  in  each  case  ? 

SOLUTIONS. 

95.  The  term  solution  has  a  wide  range  of  meaning,  embracing  phe- 
nomena   attended  by  violent  chemical    and  thermal    action    (solution  ol 
zinc  in  H^SO^  of  H^SO^  in  water),  and  also  cases  where  the  components 
become  incorporated  together  with  little  or  none  of  these  effects    (sugar 
in  water).      There  is  a  clear  distinction,  however,  between  solutions  and 
mechanical  mixtures.     The  latter  can  be  separated  by  mechanical  means; 
the  former  cannot.     Very  finely  divided  particles  ca*n  remain  suspended 
indefinitely  in  liquids,  as  in  the  case  of  emiilsions,  without  really  being  in 
solution  (cream  in  milk,  chocolate  in  water).        In  such  cases  the  com- 
ponents  may  be  separated    by    centrifugal    action,    for  example,  as  in 
separating  cream.     This  would  be  impossible  with  a  solution. 

96.  Solubility. — The   relations  between  liquids  and  solids  which 
render  solution  possible  in  some  cases  and  not  in  others,  are  not  clearly 
understood ;    but   it   must   in   general  depend   upon    relative    molecular 
attractions — the  same  sort  of  thing  which  determines  whether  a  liquid  wets 
or  does  not  wet  a  solid — the  predominance  of  adhesion  or  of  cohesion.      In 
general,  work  is  either  done  by  or  upon  the  molecules  of  the  substance 
dissolved,  resulting  in  either  cooling  or  heating.      A  liquid  at  a  given  tem- 
perature can  generally  dissolve  only  a  limited  portion  of  a  substance.     The 


SOLUBILITY DIFFUSION.  55 

quantity  usually  increases  with  the  temperature,  but  there  are  exceptions — 
sulphate  of  soda  diminishes  in  solubility  from  33°  to  120°  and  calcium  sul- 
phate above  40°.  Common  salt  has  practically  the  same  solubility  at  all 
temperatures.  Some  substances  have  no  definite  point  of  saturation,  but 
dissolve  in  all  proportions,  as  glue  in  water.  Solutions  of  liquids  in  liquids 
also  occur.  Water  mixes  with  alcohol  in  all  proportions.  The  conduct 
of  sulphuric  ether  is  peculiar;  Water  will  dissolve  about  3  per  cent  of 
ether,  or  ether  about  3  per  cent  of  water.  Beyond  this  they  will  not  mix, 
but  remain  in  separate  layers. 

97.  The  volume  of  a  solution  is  not  always  equal  to  the  sum  of  the 
volumes  of  its  constituents.     Generally  there  is  contraction,  as  in  the  case 
of  alcohol  and  water,  salt  and  water.     In  the  case  of  a  few  substances,  such 
as  ammonium  salts,  there  is  no  contraction. 

98.  Diffusion  of  solids  and  liquids  into  liquids  was  first  carefully 
studied  by  Graham  about  1850.     Any  substance  which  is  soluble  in  a  liquid, 
will,  if  left  to  itself,  gradually  diffuse  until  it  is  of  uniform  density  throughout 
— just  as  in  the  case  of  gases,  but  far  more  slowly.     The  rate  of  diffusion 
between  two  regions  of  a  liquid  is  directly  proportional  to  the  difference 
of  concentration  between  them,   and  inversely  to  their  distance,  and  is 
accelerated  by  rise  of  temperature,  (another  instance  of  increased  mole 
cular  mobility  accompanying  rise  of  temperature).     Stirring  promotes  dif- 
fusion and  uniform  mixture  by  bringing  the  masses  of  different  concentra- 
tion nearer  together.     The  effect  of  gravitation  seems  vanishingly  small  in 
retarding  diffusion  of  heavy  molecules   (e.  g.,  CuSO4  in  water  becomes 
of  uniform  concentration  throughout) .     Heavier  molecules  as  a  rule  diffuse 
more  slowly.      The  differences  of  concentration  in  a  liquid  gives  rise  to 
diffusion  convection  currents  similar  to  those  caused  by  heat. 

99.  Osmosis. — This  name  is  applied  to  the  diffusion  of  a  substance 
through  a  membrane  or  through  porous  walls.     Since  the  rate  varies  with 
different  substances,  phenomena  similar  to  those  of  gaseous  diffusion  are 
observed.     A  rubber  membrane  between  alcohol  and  water  allows  the  first 
to  pass,  but  not  the  other.     In  an  animal  membrane  the  reverse  is  true. 
If  a  porous  vessel,  such  as  an  earthenware  cup,  is  filled  with  a  concentrated 
solution  of  any  salt  and  the  vessel  immersed  in  water,  the  latter  will  pass 
through  more  readily  than  the  molecules  of  salt ;  consequently  there  is  an 
increase  of  the  contents  of  the  vessel,  which  will  produce  a  rise  of  the  solu- 
tion in  a  vertical  tube.     The  force  causing  this  is  called  osmotic  pressure. 
Pfeffer,  ofLeipsic,  in  1877,  found  that  the  osmotic  pressure  for  small  con- 
centrations increases  directly  as  the  concentration  and  the  absolute  tem- 
perature— /.   e.,   the  relations  of  density,   pressure,  and  temperature  are 
those  expressed  by  Boyle's  and  Charles'  laws.     In  order  to  isolate  the  effect 
due  to  the  salt  from  that  due  to  the  solvent,  Pfeffer  used  what  are  called 
"semi-permeable  membranes,"  made  by  precipitating  a  layer  of  ferrocy- 
anide  of  copper  in  the  walls  of  porous  cups.     This  substance  allows  water 
to  pass  freely,  but  is  impermeable  to  such  substances  as  metallic  salts, 
sugar,  etc.     By  connecting  the  vertical  tube  with  a  mercury  manometer 
Pfeffer  measured  the  osmotic  pressures  due  to  a  number  of  substances. 
Another  important  fact  discovered  by  him  was  that  solutions  with  concen- 
trations proportional  to  the  molecular  weights  of    the  substances   used 
(equimolecular  solutions)   exerted  the  same  osmotic   pressures,  showing 
that  this  pressure  depends  solely  on  the  number  of  molecules  present,  not 
their  nature.     This  is  analogous  to  Avogadros'  law.     De  Vries  found  that 


66  PROPERTIES    OF    MATTER.      . 

certain  plant  cells  contain  protoplasm  surrounded  by  a  semi-permeable 
membrane.  If  placed  in  a  solution  of  osmotic  pressure  less  than  or  equal  to 
that  of  the  contents  of  the  cell,  there  is  no  change.  If  the  solution  is  more 
concentrated,  water  is  withdrawn  from  the  cell,  the  membranous  walls  of 
which  contract.  Bonders  and  Hamburger  showed  that  the  same  is  true  of 
blood  corpuscles,  so  that  these  organic  cells  have  been  found  useful  in 
experimentation,  especially  in  reducing  solutions  to  equal  osmotic  pressure. 

100.  Dialysis. — Some  substances,   such  as  glue  and  jelly,  scarcely 
diffuse  at  all  through  porous  walls.     These  are  called  colloids.     Crystal- 
loids, or  substances  having  a  crystalline  structure,  diffuse  readily  through 
most  porous  membranes.     Mixtures  of  two  such  substances  may  then  be 
separated  by  diffusion,  which  process  is  called  dialysis. 

101.  Theory  of  Solution. — The  resemblance  between  some  of  the 
properties  of  substances  in  solution  and  those  of  gases  suggests  a  physical 
analogy  between  the  two  states.     It  is  an  experimental  fact  that  (with  the 
exceptions  noted  in  section  103)  a  substance  in  solution  exerts  an  osmotic 
pressure  equal  to  the  pressure  that  the  same  mass  would  exert  as  a  gas  if 
it  occupied  the  same  volume  at  the  same  temperature.     Van't  Hoff,  now 
a  professor  at  Berlin,  in  1887  suggested  the  hypothesis  that  the  molecules 
of  a  substance  in  solution  act  like  those  of  a  gas  in  space.     This  hypoth- 
esis has  much  to  recommend  it,  and  has  been  generally  accepted.     If  the 
analogy  were  perfect,  the  only  function  of  the  solvent  would  be  to  serve 
as  the  medium  in  which  the  molecules  may  have  free  motion.     Since  salts 
are  not  equally  soluble  in  all  liquids,  it  seems  evident,  however,  that  the 
solvent  takes  a  more  active  part,  modifying  the  phenomenon  in  a  manner 
not  yet  understood,  so  that  we  must  be  cautious  in  applying  the  laws  of 
gases  to  solutions,  except  in  a  purely  formal  way. 

102.  We  may  imagine  an  experiment  with  gases  analogous  to  Pfeffer's 
experiments  with  semi-permeable  membranes,  and  making  the  results  some- 
what clearer.     A  closed  palladium   or  platinum  cylinder  filled  with  nitro- 
gen is  placed  in  an  atmosphere  of  hydrogen  at  the  same  pressure.     The 
hydrogen  freely  enters  the  cylinder  until  its  own  partial  pressure  is  equal 
to  that  outside  (Dalton's  law).    The  nitrogen  cannot  escape,  and  so  the 
pressure  rises  on  the  inside. 

103.  There  are  many  cases  of  abnormal  osmotic  pressures  which 
may  be  explained  as  the  result  of  molecular  dissociation,  each  part  of  a 
dissociated  molecule  being  as  effective  in  producing  pressure  as  an  entire 
molecule.     This  explanation  is  borne  out  by  similar  abnormal  conduct  in 
the  raising  of  the  boiling-point  and  lowering  of  the  freezing-point  of  solu- 
tions,  and  in  electrical  conductivity.     All  of  these  phenomena  indicate 
that  metallic  acids,  bases,  and  salts  are  in  part  dissociated  in  solution  (all 
electrolytic  conductors),  while  organic  solutions  (non-conductors)  contain 
no  dissociated  elements.     Many  salts  appear  to  be  completely  dissociated 
in  very  dilute  solutions. 

Osmotic  diffusion  probably  plays  a  large  part  in  plant  and  animal 
circulation  and  assimilation.  Poynting  suggests  that  osmotic  pressure 
may  be  due  to  the  loading  of  water  molecules  by  combination  with  those 
of  a  salt,  making  it  impossible  for  those  complex  physical  molecules  to 
escape,  while  the  ordinary  water  molecules  can  enter. 


References. — Jones,  the  Modern  Theory  of  Solution,  and  Physical  Chemistry; 
Ostwald,  Solutions;  Whetham,  Solution  and  Electrolysis. 


FLOW — DIFFUSION.  57 

QUESTIONS  AND  PROBLEMS. 

59.  Five  grams  of  sugar  are  dissolved  in  one  liter  of  water.     What  is  the 
osmotic  pressure  ? 

60.  Ten  grams  of  NaCl  are  dissolved  in  water,  and  one-half  becomes  disso- 
ciated.    What  is  the  osmotic  pressure  ? 

61.  In  what  case  would  the  centrifugal  machine  fail  to  separate  the  compo- 
nents of  a  mechanical  mixture  ? 

SOLIDS. 

104.  A  solid  may  be  defined  as  a  substance  that  is  more  or  less 
rigid — i.  e. ,  its  molecules  occupy  definite  equilibrium  positions.     No  solid 
is  perfectly  rigid;  all  yield  more  or  less  to  a  deforming  force.     We  assume 
that  the  permanent  shape  of  a  solid  is  due  to  cohesive  forces  between  very 
complex  molecules.      If  two  very  smooth  and  clean  plane  surfaces  of  glass 
or  metal  be  brought  in  close  contact,  these  forces  will  hold  the  surfaces 
together  even  with  a  thin  air  film  between  them  (as  shown  by  the  "New- 
ton's rings"  between  two  glass  surfaces).     It  is  possible  that  adhesion 
between  air  and  glass  may  play  some  part  in  this  result.     Still  more 
familiar  are  examples  of  adhesion,  as  shown  by  pencil  marks  on  paper, 
gilding,  cements,  etc.      Spring  has  shown  that  powdered  bismuth  under 
6,000  atmospheres  pressure  becomes  a  crystalline  solid;  graphite  solders 
in  a  solid  mass  at  5,500  atmospheres ;  copper  filings  mixed  with  powdered 
sulphur   under   great   pressure   form    solid    and    homogeneous   cuprous 
sulphide. 

105.  Flow. — As  Tresca  has  shown,  many  solids,  such  as  the  metals, 
may  be  made  to  spread  laterally  or  to  flow  through  openings  by  applying 
sufficient  pressure.     This   phenomenon   occurs   in   wire-drawing,   in  the 
manufacture  of  metal  tubes,  and  in  coinage. 

106.  Diffusion. — There  are  instances  of  the  diffusion  of  one  solid 
in  another.     Carbon  diffuses  through  iron  in  the  cementation  process  of 
making  steel.     When  metals  are  welded  at  high  temperatures  there  is 
some  interdiffusion  at  their  common  boundary.     Mixtures  of  metals  in  the 
form  of  powder  have  been  known  to  form  alloys  when  subjected  to  pres- 
sures of  several  thousand  atmospheres. 

Joseph  Henry  showed  that  a  leaden  U-rod  with  one  limb  in  a  vessel  of 
mercury  acted  as  a  siphon.  The  mercury  diffused  through  the  rod  and 
dropped  from  the  lower  end  of  the  outer  limb  until  the  mercury  in  the 
vessel  was  exhausted.  He  also  showed  that  on  heating  silver-plated  copper 
the  silver  will  disappear  beneath  the  surface,  but  may  be  again  exposed 
on  dissolving  the  surface  copper  in  acid. 

Roberts-  Austen^  showed  that  if  a  lead  cylinder  7  cm.  long  was  placed 
on  a  piece  of  gold,  with  close  contact,  the  gold  diffused  rapidly  into  the 
lead  at  high  temperatures  (but  below  the  melting  point  of  lead).  At  a 
temperature  of  251°  C.  the  gold  penetrated  to  the  top  of  the  cylinder  in 
three  days.  Traces  of  such  diffusion  were  observed  even  at  ordinary 
temperatures.  The  lead  was  left  in  contact  with  the  gold  for  four  years, 
when  traces  of  gold  were  found  in  four  parallel  slices,  each  .75  mm.  thick, 
cut  from  the  bottom  of  the  cylinder.  Such  results  indicate  the  existence 
of  translatory  molecular  motions  even  in  solids,  and  give  us  reason  to 
suspect  that  there  may  be  constant  diffusion  of  gold  within  itself.  Roberts- 
Austen  also  found  some  diffusion  of  gold  into  silver  and  copper  at  higher 
temperatures. 

(1)  Nature,  Vol.  54,  p.  55,  1896;  Proc.  Royal  Soc.,  67,  1900. 


58  PROPERTIES     OF     MATTER. 

107.  Solution.  —  We  might  say  in  the  above  case  that  the  gold  was 
dissolved  by  the  lead.      Selective  solubility  is  shown   in  many  cases  by 
melted  metals.     Mercury  will  dissolve  copper,   but  not  iron.     Melted  lead 
will  dissolve  tin  but  not  zinc.      Alloys  of  the  metals  in  many  cases  resembles 
solutions  in  their  properties.     In  some  cases  these  solid  solutions  may  be 
saturated,  that  is,  one  component  will  not  alloy  with  more  than  a  certain 
proportion  of  another.     In  all  these  cases  we  see  how  molecular  mobility 
is  increased  by  heat. 

MOLECULAR  FORCES  AND  STRUCTURE. 

The  properties  of  solids  depend  on  the  nature  of  the  molecular  con- 
straint, which  determine  their  rigidity,  hardness,  ductility,  etc.  The 
molecular  force  which  tends  to  make  molecules  resume  their  equilibrium 
positions  after  being  displaced  is  called  elasticity. 

108.  Elasticity.  —  In  order  that  a  body  may  be  elastic  it  must  have 
the  properties  of  (1)  resistance  to  a  distorting  force;  (2)  a  force  of  restitu- 
tion (resilience),  which  will  restore  it  to  its  original  position  on  removing 
the  disturbing  force.     In  perfectly  elastic  bodies,  such  as  fluids,   the  force 
of  restitution  is  equal  to  the  distorting  force. 

In  bodies  having  a  crystalline  structure,  elasticity  and  other  properties 
are  different  along  different  axes.  Bodies  with  properties  uniform  in  all 
directions  are  called  isotropic.  The  elastic  relations  of  crystalline 
(aelotropic)  bodies  are  very  complicated  and  cannot  be  discussed  here,  but 
in  the  case  of  isotropic  bodies  they  depend  upon  two  elements  only;  the 
resistance  to  compression  and  the  resistance  to  change  of  form. 

The  measiire  of  elasticity  is  the  ratio  stress  /  strain.  The  coefficient 
of  volume  elasticity  or  of  compressibility  is  — 

i       stress  per  unit  area          pvo 

K  =  —  -  -  —  -  -  - 


—  -  ;  -  ;  —  -  r-  •=     -  r 

strain   per  unit   vol.      (&  —  v  ) 

The  resistance  offered  to  change  of  shape  is  called^  rigidity.  This  is 
the  force  called  out  by  the  torsion  of  a  wire.  In  cases  where  there  is  no 
change  of  volume  a  simple  shear  is  produced,  similar  to  the  effect  produced 
by  applying  a  tangential  pressure  to  the  top  of  a  pack  of  cards.  The  strain 
is  measured  by  the  angular  shear,  which  for  small  deformations  is  b  /  a, 
the  tangent  of  the  angle  of  twist  between  two  planes  1  cm.  apart.  The 
coefficient  of  rigidity  is  defined  as 

(99)  n  =pa  /  b, 

p  being  the  tangential  force  per  unit  area. 

As  a  rule,  the  compressibility  of  solids  is  less  than  their  rigidity.  Cork 
is  an  exception. 

In  cases  of  longitudinal  compression  or  elongation  the  coefficient  of 
elasticity  used  is  Young's  modulus: 

nnm  Jl/r       stress  per  unit  area          PL 

(    A.\J\J  )  J.YL  --        "  .  ~.          .  7~    ~~77~   --  1          * 

strain  per    unit    length       al 

P  being  the  applied  force,  a  the  cross-section  of  the  body,  L  its  length, 
and  /  the  elongation  or  compression.  M  may  be  thus  defined  as  the 
force  per  unit  area  required  to  double  the  length  of  a  given  substance, 
assuming  the  law  of  stretching  to  hold  indefinitely. 


ELASTICITY — HARDNESS.  59 

Elongation  in  general  involves  lateral  contraction.  It  may  be  seen 
that  an  imaginary  cube  in  the  substance  having  the  diagonals  respectively 
parallel  and  perpendicular  to  the  direction  of  elongation  will  have  one 
diagonal  lengthened  and  the  other  shortened,  equivalent  to  a  shear  and  a 
rotation  Consequently  Young's  modulus  is  a  function  of  both  the  com- 
pressibility and  the  rigidity  of  the  substance. 

The  three  elastic  coefficients  k,  n,  and  J/are  usually  very  large  quanti- 
ties. Their  values  for  some  substances  are: 

k  n  M 


Steel 18X10"  8x10"        20x10" 

Brass 10x10"        3.5x10"      9.5X10" 

Glass 3.5X10"        2.2x10"         6x10" 

The  three  coefficients  may  be  respectively  defined  as  the  force  in  dynes 
required  to  reduce  a  unit  volume  to  zero,  to  shear  a  unit  cube  45°,  and  to 
double  the  length  of  a  unit  cube. 

In  the  bending  of  rods,  Young's  modulus  is  the  coefficient  involved; 
in  the  torsion  of  rods  or  wires,  it  is  the  coefficient  of  rigidity. 

109.  Limit  of  Elasticity. — If  bodies  are  strained  bevond  their  elastic 
limit,  there  is  a  permanent  set  produced.     When  a  solid  is  strained  for  the 
first  time,  there  is  a  small  permanent  set  even  for  the  smallest  distorsions; 
for  future  strains  not  exceeding  that  producing  the  set,    Hooke's  law  is 
almost  perfectly  obeyed.     Many  metals  have  a  crystalline  structure,  and 
the  limit  seems  to  be  reached  when  the  crystals  begin  to  slip. 

110.  Hooke'  s  Law. — In  most  cases  it  is  found  that  the  deformations  of 
elastic   bodies   are   proportional  to  the  forces   producing   them.     Exact 
measurements  show  that  this  is  not  strictly  true  for  large  deformations, 
but  the  law  is  nearly  enough  true  in  most  cases  to  be  made  the  basis  of 
numerical  calculations. 

Increased  temperature  usually  diminishes  elasticity — another  example 
of  the  effect  of  heat  in  breaking  down  molecular  constraints. 

111.  Elastic  Fatigue. — When  a  body  has  been  subjected  to  rapidly 
alternating  strains  for  some  time,    elastic  fatigue  occurs;  for  example,   a 
tuning-fork  which  has  been  maintained   in   continuous  vibration   seems  to 
lose  a  considerable  portion  of  its  elasticity.     Kelvin  found  that  a  torsion 
pendulum  which  diminished  its  amplitude  of  vibration  by  one-half  in  100 
vibrations  would,  after  fatigue  set  in,  drop  to  one-half  its  original  amplitude 
in  45  vibrations.     If  the  wire  of  such  a  pendulum  be  kept  twisted  to  the 
right  for  some  time,  and  then  to  the  left  for  the  same  time,  on  releasing  it 
the  last  twist  will  be  first  undone,  and  then  the  first  (after-affecf). 

A  perfectly  elastic  body  set  in  vibration  would  continue  to  vibrate 
indefinitely.  All  the  work  spent  in  producing  the  displacement  would  be 
converted  into  potential  energy  of  the  molecules.  In  all  actual  cases 
some  of  the  work  is  expended  in  overcoming  molecular  friction  or  viscosity 
at  each  vibration ;  so  that  there  is  constant  expenditure  of  energy,  or  rather 
its  transformation  to  heat,  and  the  vibration  ceases. 

Such  properties  as  hardness,  tenacity,  ductility,  etc.,  depend  on  the 
nature  of  the  molecular  constraints. 

112.  Hardness. — Below  is  Mohs'  scale  of  hardness,  each  substance 
being  able  to  scratch  all  placed  before  it  on  the  scale  :  1,  talc  ;  2,  gyp- 
sum ;    3,    calcite  ;    4,    fluorite  ;    5,    apatite  ;    6,    feldspar  :    7,   quartz  ;  8, 
topaz  ;  9,  sapphire  ;   10,  diamond. 


60  PROPERTIES    OF    MATTER. 

V-^. 

A  rapidly  moving  body  is  in  some  cases  able  to  cut  bodies  harder  than 
itself.  A  soft-iron  wheel  revolving  at  a  great  speed  can  cut  steel  or 
even  quartz.  In  such  cases  there  is  mutual  wear,  but  the  action  is  con- 
centrated on  the  hard  object  and  distributed  over  the  other.  Similar 
effects  are  seen  in  the  action  of  a  sand  blast,  and  the  erosion  of  rocks  by 
water.  This  erosion  is  due  to  suspended  particles  of  sand.  In  Sicily  a 
deposit  of  lava  made  by  Mt.  Etna  in  1603  had  a  gorge  30  feet  deep  and 
several  hundred  feet  wide  cut  through  it  by  a  stream  in  little  more  than 
two  centuries. 

113.  Tempering   and  Annealing. — Some   metals,    such    as    steel, 
when  heated  and  cooled  suddenly  by  immersion  in  a  liquid  are  rendered 
much  harder.     In  case  of  steel  the  resulting  density  is  less  than  before. 
In  one  case  the  density  was  7.817  before  tempering  and  7.743  afterwards. 
On  reheating  it  returned  to  nearly  its  original  density.      In  brass  and 
alloys  containing  tin  there  is  an  increase  of  density.     Bronze  is  tempered 
in  a   way  the  reverse  of  that  applied  to  steel  ;  it  is  softened  by  rapid 
cooling. 

Annealing  is  the  name  applied  to  the  tempering  of  such  substances  as 
glass.  Glass  is  made  much  tougher  by  slow  and  uniform  cooling.  If  the 
surface  is  cooled  rapidly,  the  subsequent  contraction  of  the  interior  pro- 
duces an  interior  tension  causing  an  unstable  state,  as  shown  in  Prince 
Rupert  drops,  made  by  dropping  melted  glass  in  water.  Such  bodies 
will  stand  a  severe  blow  or  pressure,  but  burst  into  powder  when  the  stem 
is  broken  off.  Barus  has  shown  that  the  unstable  condition  is  relieved  by 
dissolving  the  external  layers  in  hydrofluoric  acid. 

114.  Tenacity    is   the   resistance   to   fracture    due    to  elongation. 
Fracture  implies  a  lack  of  homogeneity,   since  a  perfectly  uniform  body 
subjected  to  stress  would  rupture  at  all  points  simultaneously — /.  ^.,  fall 
to  powder.     Steel  is  the  most  tenacious  and  lead  the  least  tenacious  of  the 
metals.     Toughness  implies  tenacity  rather  than  hardness.     The  tenacity 
of  wires  is  increased  by  the  process  of  drawing,  so  that  a  very  small  wire 
will  support  proportionally  twice  as  much  as  a  large  one. 

115.  Friction  is  the  resistance  offered  to  the  motion  of  one  body 
over  another,   due  to  the  interlocking  irregularities  of  their  surfaces.     It 
is  not  a  force  in  the  most  general  sense,  since  it  can  only  destroy,  never 
originate,   motion.     Experiment  shows  that  the  resistance  due  to  friction 
varies  with  the  substances  tested  and  with  the  state  of  their  surfaces,  as 
first  shown  by  Coulomb.      It  is  approximately  proportional  to  the  normal 
pressure  between  them  and  approximately  independent  of  the  area  of  con- 
tact and  velocity  of  sliding.     The  coefficient  of  friction  between  two  sub- 
stances is  the  ratio  of  the  tangential  force   of  friction  and  the  normal 
pressure. 

(101)  *  =  £ 

If,  for  example  it  requires  a  force  of  two  pounds  weight  to  pull  a 
weight  of  10  pounds  along  a  level  surface,  /*  =  .2.  Some  values  of  the 
coefficient  of  friction  are:  Oak  on  oak,  .41;  wagon  on  ordinary  road, 
.04;  iron  on  iron  rails,  .004.  Rolling  friction  (last  two  cases)  is  much 
less  than  sliding  friction.  The  fact  that  the  total  force  of  friction  depends 
upon  the  pressure  of  the  surfaces  in  contact  is  illustrated  by  the  great 
firmness  of  grip  of  a  rope  wrapped  only  once  or  twice  around  a  tree  or 


CRYSTALS MOLECULAR    THEORIES.  61 

post.  As  shown  in  discussing  surface  tension,  the  normal  pressure  of  a 
curved  surface  is  inversely  proportional  to  the  radius  of  curvature — hence 
the  great  resistance  to  slipping  when  the  tree  or  post  has  a  small  radius  of 
curvature  ;  even  when  it  is  smooth. 

116.  Crystals. —  Nearly  all  inorganic  substances  have  in  their  natu- 
ral state  a  crystalline  structure.  Crystals  are  bounded  by  plane  faces 
symmetrically  arranged  with  reference  to  three  or  four  diametral  lines 
called  axes,  and  in  a  given  species  the  angles  of  inclination  between  the 
faces  is  constant. 

Differences  in  length  of  axes  are  accompanied  by  differences  of 
physical  properties,  so  that,  except  in  the  case  of  crystals  belonging  to 
the  isometric  system,  in  which  all  the  axes  are  equal  (such  as  common 
salt),  there  are  differences  along  different  axes  in  hardness,  elasticity, 
thermal  expansion  and  conductivity,  and  optical  effects. 

Crystalization  is  favored  by  changes  of  state,  such  as  passage  from 
gaseous  to  solid  state  (phosphorus  volatilized  in  a  vacuum  at  low  tem- 
perature, sublimation  of  arsenic  at  dull  red  heat),  solidification  from 
fusion  (bismuth),  evaporation  of  solution  of  salt. 

In  all  supersaturated  solutions  of  salt,  crystallization  is  started  by 
the  addition  of  a  small  crystal  of  the  same  form,  which  serves  as  a 
nucleus.  A  broken  crystal  placed  in  a  solution  of  the  substance  of 
which  it  is  composed  will  repair  the  flaw  before  general  growth  takes 
place. 

It  is  difficult  to  explain  the  conduct  of  crystals  except  by  assuming  a 
very  regular  geometrical  arrangement  of  their  molecules,  tending  to  pro- 
duce stability  under  the  action  of  molecular  forces. 

Many  organic  compounds  have  the  same  chemical  constitutions  (are 
isomeric),  but  very  different  physical  properties.  This  can  be  explained 
only  as  a  result  of  different  arrangement  of  atoms  in  a  simple  molecule,  or 
different  molecular  aggregations. 

117.  Molecular  Theories. — There  have  been  various  hypotheses 
as  to  the  nature  of  molecules,  chiefly  the  following :  Lucretius,  the  Roman 
poet,  believed  that  the  atoms  of  which  they  are  composed  are  hard, 
indivisible  bodies,  whose  properties  depend  on  their  shape.  For  instance, 
those  of  honey  and  milk  are  round  and  smooth,  while  those  of  disagree- 
able substances  are  rough  and  hooked.  Boscovich  held  that  atoms  are 
merely  centers  of  force.  Sir  William  Thomson  (Lord  Kelvin)  advanced 
the  hypothesis  that  each  atom  is  a  little  vortex  in  a  perfect  fluid — the 
ether  that  transmits  the  waves  which  produce  the  effects  of  light  and  heat. 
Helmholtz  showed  that  vortices  in  such  a  fluid — that  is,  one  which  is  per- 
fectly elastic  and  non-viscous — could  neither  be  originated  nor  destroyed 
by  any  means  known  to  us.  A  special  creative  act  would  be  necessary. 
Many  of  the  properties  of  such  vortices  are  illustrated  in  a  crude  way  by 
the  conduct  of  smoke  rings.  The  equilibrium  form  of  such  rings  is  a 
circle.  If  distorted,  they  oscillate  about  this  form.  They  are  indivisible 
because  it  is  impossible  to  touch  them.  They  rebound  from  each  other 
and  from  other  bodies  with  perfect  elasticity.  Two  rings  which  approach 
each  other  under  certain  conditions  will  not  separate,  but  will  move  on 
together,  rotating  about  each  other,  suggesting  chemical  combination. 
According  to  this  hypothesis,  all  matter  is  fundamentally  the  same,  the 
differences  of  the  elements  depending  on  the  size  or  complexity  of  the 
vortex  atoms,  which  may  consist,  in  many  cases,  of  a  number  of  rings 
inseparably  linked  together. 


62  PROPERTIES    OF    MATTER. 

These  speculations  are  interesting,  but  no  theory  has  yet  been  found 
which  is  self-consistent  and  satisfactory  in  all  respects. 

118.  Size  of  Molecules. — There  appears  to  be  no  possible  means 
of  ascertaining  the  size  of  a  molecule,  but  some  physical  phenomena  give 
us  a  rough  idea  of  the  maximum  distance  between  their  centers  or  the 
diameter  of  their  spheres  of  activity. 

The  thickness  of  the  film  of  a  soap  bubble  may  be  determined  with 
considerable  accuracy  by  the  colors  due  to  interference  of  light  waves  of 
known  length  reflected  from  its  outer  and  inner  surfaces.  Reinhold  and 
Riicker  have  shown  in  this  manner  that  a  minimum  value  of  the  surface 
tension  is  reached  when  the  bubble  is  .0000012  cm.  thick.  It  is  not  prob- 
able that  such  a  film  is  many  molecules  thick. 

Quincke,  by  observing  the  angle  of  contact  between  liquids  and  glass 
coated  with  wedge-shaped  films  of  silver,  observed  that  the  effect  of  the 
glass  upon  the  contact  angle  began  to  be  exerted  through  the  film  when 
its  thickness  (determined  by  color  phenomena)  was  about  .0000050  cm. 

Wiener  has  been  able  to  detect  optically  the  existence  of  a  film  of 
silver  not  more  than  .00000002  cm.  thick  deposited  on  mica. 

By  extending  a  cubic  centimetre  of  water  into  a  film  of  .00000001  cm. 
thickness  it  may  be  shown  that  the  heat  necessary  to  be  supplied  to  main- 
tain a  constant  temperature  will  be  sufficient  to  vaporize  the  water,  or  at 
this  thickness  the  molecular  forces  begin  to  break  down. 

These  figures  give  some  idea  of  the  maximum  limit  of  range  of  molec- 
ular forces. 

119.  Free  Path. — As  the  rarefaction  of  a  gas  increases,  the  length 
of  the  mean  free  path,  or  the  distance  the  molecules  can  move  without 
collision,  is  increased.      By  applying  the  kinetic  theory  of  gases  to  the 
phenomena  of  diffusion  and  viscosity,   this  distance  may  be  determined. 
In  air  at  ordinary  pressures  it  is  about  .00001  cm.     In  a  good  vacuum 
it  may  be  several  inches.     The  effects  observed  in  Crookes'  radiometer 
are  due  to  increased  length  of  path.     A  little  windmill,  with  vanes  of  mica 
or  aluminum,  is  mounted  on  an  axis  in  a  very  low  vacuum.     One  side  of 
the  vanes  is  blackened,  and  on  exposing  the  instrument  to  a  source  of 
heat  the  blackened  side  becomes  wrarmer  than  the  other,   owing   to  its 
greater  absorptive  power.     The  molecules  which  strike  on  that  side  have 
their  kinetic  energy  increased  and  rebound  with  greater  velocity  than  those 
on  the  other  side.     The  unbalanced  reaction  causes  rotation. 


References. — Maxwell,   article   "Atom,"  Ency.   Brit.;  Wm.  Thomson,   Pop. 
Lectures,  Vol.  I— Size  of  Atoms;  Tait,  Properties  of  Matter. 

QUESTIONS. 

62.  A  piece  of  glass  fastened  at  one  end  or  a  dead  tamarack  limb  will  fly  in 
several  pieces  if  broken  by  pulling  on  free  end.     Explain. 

63.  A  stretched  horizontal  wire  will  not  support  as  heavy  a  weight  at  its 
center  without  breaking  as  it  would  if  hung  vertically.     Explain. 

64.  What  kind  of  knots  in  ropes  hold  most  firmly  without  slipping  ? 

65.  Explain  the  amalgamation  process  of  collecting  gold  from  crushed  quartz. 


HEAT. 


120.  When    we  touch  bodies  we   generally  become   aware  of  two 
distinct   sensations — one   of  material   resistance,    or   force,    and   one   of 
hotness  or  coldness.     To  the  latter  sensation  we  give  the  name  of  heat  or 
cold.     We  also  give  the  name  of  heat  to  the  cause  of  these  sensations, 
regarding  cold  not  as  a  positive  property,   but  merely  as  the  absence  of 
heat.     The  same  confusion  exists  in  the  use  of  the  words  sound  and  light, 
which  are  applied  both  to  the  sensations  and   to  the  agents  producing 
them.     In  the  study  of  physics,  however,  we  shall  concern  ourselves  only 
with  the  nature  of  the  agencies  producing  these  sensations. 

121.  Production  of  Heat. — Experience  shows  that  heat  may  be 
produced  in  several  ways: 

By  chemical  action,  as  in  combustion. 

By  change  of  state,  as  in  the  condensation  or  freezing  of  water. 

}$y  friction.  Savages  produce  fire  by  friction  between  two  pieces  of 
wood.  When  a  copper  or  lead  wire  is  bent  rapidly  back  and  forth  heat 
is  produced  by  molecular  friction. 

By  compression  or  percussion,  as  by  suddenly  compressing  air  or 
hammering  a  piece  of  metal. 

By  an  electric  current  in  overcoming  resistance,  as  in  an  incandescent 
lamp.  In  all  these  cases  except  perhaps  the  last,  we  may  imagine  a  clash- 
ing of  molecules,  increasing  their  kinetic  energy. 

122.  Effects    of   Heat. — When   bodies  are  heated    the   following 
effects  may  be  produced: 

Change  of  hotness,  or  of  temperature. 

Change  of  volume,  as  in  the  case  of  the  thermometer. 

Change  of  state,  as  in  melting  of  ice  and  evaporation  of  water. 

Change  in  molecular  constraints,  shown  by  change  in  elasticity  and 
viscosity,  in  magnetic  strength. 

Increased  molecular  velocities,  as  in  diffusion  of  gases  and  liquids  and 
in  evaporation. 

Chemical  dissociation. 

Production  of  electric  current,  in  thermal  couple. 

123.  Nature  of  Heat. — A  Form  of  Energy. — If  a  hot  body  is  in 
contact  with  a  colder  one,  we  find  that  the  colder  body  becomes  hotter 
and  the  hot  one  colder  until  both  come  to  the  same  state.     Something 
must   pass   from  one  to   the  other  to   produce  this  change:     There  are 
apparently  only  two  things  in  the   physical  world  which  appeal  to    our 
senses — matter  and  energy.     To  which  of  these  classes  does  heat  belong? 
is  a  question  which  can  only  be  decided  by  a  careful  study  of  the  effects 
and   method  of  production  of  heat,  and  its  relation   to   other   physical 
phenomena. 

Many  of  the  ancient  Greeks  believed  in  the  existence  of  self-repellent 
fire  atoms,  which  caused  expansion  by  forcing  themselves  between  the 
atoms  of  matter.  Modifications  of  this  theory  held  their  ground  until  the 
beginning  of  the  nineteenth  century.  At  that  time  the  generally  accepted 
view  was  that  heat  or  caloric  is  a  subtle,  elastic,  weightless  fluid,  whose 


64  HEAT. 

parts  are  self-repellent,  thus  causing  expansion,  conduction,  and  radiation. 
This  idea  was  strengthened  by  the  discovery  of  the  fixed  gases,  which 
appeared  to  have  many  properties  in  common  with  this  hypothetical  fluid. 
Opposed  to  this  material  theory  of  heat  is  the  dynamical,  which  assumes 
it  to  be  the  effect  of  rapid  motion  in  the  particles  of  matter  (and  conse- 
quently a  form  of  energy).  This  idea  has  had  some  advocates  from  the 
earliest  days  of  Greek  speculation.  Lord  Bacon,  after  considering  the 
effects  of  heat,  concluded  that  it  must  be  due  to  motion.  Newton  held  a 
similar  opinion.  All  this  was,  however,  scarcely  more  than  speculation, 
and  no  definite  experiments  to  test  the  theory  were  undertaken  until  the 
end  of  the  eighteenth  century.  Count  Rumford  was  the  pioneer  in  this 
work.  His  attention  was  called  to  the  subject  in  1778  by  observing  that 
when  a  blank  cartridge  is  fired  from  a  gun,  the  barrel  is  more  heated  than 
when  a  ball  is  fired  (and  more  mechanical  work  done),  suggesting  some 
relation  between  heat  and  energy.  Later  he  showed  that  if  caloric  is  a 
material  fluid  it  has  no  weight.  In  1798,  while  boring  cannon  for  the 
king  of  Bavaria,  he  observed  that  heat  was  evolved  without  limit  as  long 
as  the  boring  continued.  He  raised  the  temperature  of  the  cannon  to  the 
boiling  point  of  water  by  boring  two  and  a  half  hours.  If,  as  the  calorists 
believed,  heat  is  a  material  fluid,  we  should  expect  some  limit  to  its 
production.  Rumford  concluded  that  ' '  anything  which  any  insulated 
body  or  system  of  bodies  can  continue  to  furnish  without  limitation  cannot 
possibly  be  a  material  substance,  and  it  appears  to  me  to  be  extremely 
difficult,  if  not  quite  impossible,  to  form  any  distinct  idea  of  anything 
capable  of  being  excited  and  communicated  in  the  manner  in  which  heat 
was  excited  and  communicated  in  these  experiments  unless  it  be  motion." 

The  calorists  explained  that  the  capacity  for  heat  of  the  iron  (the 
amount  of  heat  required  to  change  its  temperature  by  a  definite  amount) 
was  diminished  by  being  reduced  to  powder  or  borings,  thus  causing  an 
evolution  of  heat.  Rumford  showed  that  the  capacity  was  unchanged, 
but  his  proof  was  not  conclusive,  because  he  did  not  show  that  the  total 
amount  of  heat  in  the  one  case  was  equal  to  that  in  the  other,  as  he  might 
have  done  by  finding  whether  the  same  quantity  of  heat  was  required  to 
melt  equal  masses  of  the  iron  and  the  borings,  or  was  evolved  in  dissolving 
equal  amounts  in  acid,  and  thus  reducing  them  both  to  an  identical 
physical  condition.  Nevertheless,  the  explanation  of  the  calorists  fails  to 
hold  when  we  consider  the  case  when  friction  produces  heat  without 
altering  in  any  way  the  physical  condition  of  the  substance,  as  when  it  is 
evolved  by  churning  a  liquid. 

About  the  same  time  Humphrey  Davy  melted  two  pieces  of  ice,  at  an 
original  temperature  below  the  freezing  point,  by  rubbing  them  together 
in  a  receiver  exhausted  of  air  and  surrounded  by  ice.  In  this  case  the 
communication  of  heat  from  external  sources  was  prevented,  and  an 
expenditure  of  energy  alone  resulted  in  the  production  of  heat.  There 
was  no  question  of  the  capacity  of  the  substance,  for  that  of  ice  is  less  than 
that  of  water,  not  greater,  as  demanded  by  the  caloric  theory. 

In  considering  the  methods  of  production  of  heat  it  is  found  that  when- 
ever it  is  evolved  there  is  a  disappearance  of  mechanical,  chemical,  or 
electrical  energy.  If  we  impart  heat  to  matter,  we  find  that,  in  general, 
expansion  is  produced,  which  may  do  mechanical  work  against  molecular 
attraction  or  external  resistance;  or  the  molecular  constraints  are  partially 
or  completely  overcome,  as  in  melting  ice,  vaporizing  water,  or  producing 
chemical  dissociation  and  changes  in  elasticity  and  viscosity;  or  electrical 


SCALES    OF    TEMPERATURE THERMOMETERS.  65 

energy  is  developed,  as  in  the  production  of  thermal  currents;  or  an 
increase  of  pressure,  associated  with  an  increase  of  temperature,  is  imparted 
to  gases,  which  we  explain  by  the  kinetic  theory  as  the  result  of  increased 
kinetic  energy  of  the  molecules.  As  shown  by  Joule,  a  gas  is  cooled  by 
expansion  when  it  does  work  against  external  pressure,  but  not  when  it 
expands  into  a  vacuum.  The  conclusion  seems  irresistible  that  heat  is  a 
form  of  energy.  In  the  case  of  gases,  at  least,  it  seems  as  though  it  must 
be  in  the  form  of  molecular  kinetic  energy;  and  the  same  seems  to  be 
probably  true  of  solids,  as  indicated  by  the  communication  of  heat  by 
contact,  and  by  the  demagnetization  of  a  magnet  by  heat. 

The  caloric  theory  was  completely  overthrown  only  after  the  experi- 
ments of  Mayer  (who  first  distinctly  stated  the  idea  of  conservation  of 
energy  in  1842)  and  of  Joule,  who  established  a  definite  numerical  relation 
between  heat  and  energy,  known  as  the  mechanical  equivalent  of  heat. 
These  experiments  will  be  described  later. 


References. — Stewart,  Conservation  of  Energy;  Tait,  Heat;  Tyndall,  Heat  a 
Mode  of  Motion;  Tyndall,  Fragments  of  Science — Count  Rumford. 


124.  Quantity    of    Heat. — Heat  being  a  form   of  energy,    it   is 
capable  of  quantitative  determination  in  terms  of  work   done  or  other 
effects  produced.     This  subject  will  be  discussed  later. 

125.  Temperature. — Another  attribute  of  heat  which  more  directly 
affects    our   senses   than   its    quantity   is    the   quality  of  hotness,   or  its 
intensity.     To  this  quality   we  give  the  name   of  temperature,   meaning 
thereby  the  degree  of  hotness,    or  the  condition  which  determines   the 
flow   of  heat   from   one  body  to  another.      It  is  analogous  to  pressure  in 
liquids  or  electromotive  force  in  electricity,   which  determine  the  direc- 
tion of  flow  of  the  liquid  or  electric  current. 

THERMOMETRY. 

126.  Scales  of  Temperature. — We  may  say  that  one  body  is 
hotter  than  another  when  heat  will  flow  from  the  first  to  the  second;  or 
that  two  bodies  are  at  the  same  temperature  when  in  thermal  equilibrium. 
Further  than  this  there  is  no  absolute  method  of  measuring  temperatures, 
or  saying  that  one  temperature  is  so  many  times  greater  than  another,  in  the 
sense  that  we  say  that  one  length  is  so  many  times  greater  than  another. 
It  follows  that  all  scales  of  temperature  are  more  or  less  arbitrary.     Such 
scales  may  be  based  on  any  continuously  changing  effect  of  heat,  which  is 
always  the  same  under  the  same  conditions — such  as  the  series  of  colors 
assumed  by  polished  steel  when  tempered;  the  intensity  of  a  thermo-electric 
current;  the  change  of  electric  resistance  of  a  metal;  the  change  in  elasticity 
of  a  substance;  the  rate  of  evaporation  under  given  conditions,   or  the 
expansion  of  a  given  substance.     The  latter  is  the  effect  most  commonly 
employed  in  the  construction  of  thermometers.     As  a  rule,   fluids  such  as 
air  or  mercury  are  used,  owing  to  their  great  expansibility.     The  expansion 
of  the  containing  vessel  may  generally  be  neglected. 

127.  Thermometers. — One  of  the  earliest  forms  of  thermometer 
was  made  by  Galileo,  about  1592.     It  consisted  of  a  glass  bulb  with  a  long 
neck  inverted  in  and  partly  filled  with  water  or  alcohol,  which  rose  or  fell 
in  the  tube  with  changes  in  the  volume  of  air.      There  was  no  definite 
method  of  calibrating  these  early  thermometers,   except  by  marking  on 


66  HEAT. 

the  stem  the  points  corresponding  to  the  greatest  heat  of  summer  or  cold 
of  winter,  until  the  time  of  Boyle,  who  about  1665  used  the  temperature 
of  melting  ice  as  one  fixed  point;  and  Huyghens  about  the  same  time 
suggested  the  use  of  the  temperature  of  boiling  water  for  another.  This 
method  was  adopted  by  Newton.  The  intervening  space  on  the  stem  was 
divided  into  any  number  of  parts,  at  the  whim  of  the  maker,  and  the 
increments  of  temperature  were  assumed  proportional  to  the  apparent 
increments  of  volume. 

The  first  thermometers  of  modern  form  were  made  by  Fahrenheit,  of 
Holland,  about  1714.  He  was  the  first  to  note  that  the  temperature  of 
boiling  water  varies  with  the  pressure,  and  to  adopt  a  standard  barometric 
pressure — 29.8  inches  of  mercury.  The  corresponding  point  of  the  scale 
he  called  212°  ;  that  of  melting  ice  32°,  and  that  of  a  mixture  of  salam- 
moniac  and  ice,  0°. 

Reaumur,  of  France,  about  1730  made  a  thermometer  in  which  the 
point  corresponding  to  the  freezing  of  water  was  marked  0°,  and  that 
corresponding  to  its  boiling,  80°. 

Celsius,  professor  of  astronomy  in  the  University  of  Upsala,  about 
1740  made  a  mercurial  thermometer  in  which  the  scale  between  these 
points  was  divided  into  one  hundred  equal  parts,  and  the  boiling-point 
numbered  0°;  the  freezing-point,  100°.  The  modern  centigrade  ther- 
mometer, almost  exclusively  used  in  scientific  work,  is  the  same,  with  the 
scale  reversed. 

Guy  Lussac  discovered  that  the  temperature  of  boiling  water  depends 
somewhat  on  the  nature  of  the  containing  vessel,  and  it  has  also  been 
found  to  depend  on  the  amount  of  foreign  substances  in  solution.  The 
temperature  of  steam  from  water  boiling  under  a  given  pressure,  however, 
is  found  to  be  invariable,  and  the  temperature  of  melting  ice  is  practically 
independent  of  pressure. 

Construction. — The  ordinary  mercurial  thermometer  is  made  by  blow- 
ing a  bulb  on  a  small  glass  tube,  filling  the  bulb  and  part  of  the  tube  with 
mercury,  and  expelling  the  air  by  boiling  the  mercury  until  the  stem  is 
filled  with  mercury  vapor  alone.  The  tube  is  then  sealed.  The  points  at 
which  the  mercury  stands  when  the  bulb  is  placed  in  a  mixture  of  ice  and 
water  and  in  steam  from  water  boiling  under  a  pressure  of  76  centimeters 
of  mercury  are  marked,  and  the  intervening  space  divided  into  100  parts 
of  equal  volume,  called  degrees.  Since  glass  tubes  are  not  usually 
uniform,  they  must  be  calibrated  by  means  of  a  thread  of  mercury,  the 
length  of  which  is  measured  in  different  parts  of  the  tube.  It  is  to  be 
noted  that  increments  of  temperature  are  proportional  to  apparent,  not 
true,  changes  of  volume,  since  the  expansion  of  the  glass  is  ignored. 

Different  substances,  as  a  rule,  have  different  laws  of  expansion,  so 
that  thermometers  made  of  them  will  differ  in  their  readings,  except  at 
the  fixed  points.  Below  some  comparisons  are  made  of  different  thermo- 
metric  substances: 

Mercury  in  glass 0°  45°  50°  100°  150°         200° 

Sulphuric  acid 0°  41°  ...  100°  

Cadmium 0°  ...  53°.3  100°  141°.  4     179°.  1 

Silver 0°  ...  51°.7  100°  145°.3     188°.2 

It  is  found,  however  (as  expressed  by  Charles'  law),  that  thermom- 
eters made  of  different  gases  agree  almost  exactly  through  wide  ranges  of 
temperature.  "Furthermore,  it  will  be  shown  later  that  the  increase  in 


Q 


Vn 


, 

E     (__, 


• 


M 


A  Q 


o 


., 


AT 


AQ 


, 


^c     - 


llr     - 

^JL 


QUANTITY    OF    HEAT'— SPECIFIC    HEAT.  67 

volume  of  a  gas  under  given  conditions  is  almost  exactly  proportional  to 
the  amount  of  heat  absorbed  by  it.  A  gas  or  air  thermometer  is  there- 
fore adopted  as  the  most  appropriate  for  scientific  measurements.  Such  a 
thermometer  is,  however,  inconvenient  to  use.  It  is  found  that  of  all 
liquids  mercury  has  the  most  uniform  expansion  through  ordinary  ranges, 
as  shown  by  comparison  with  the  air  thermometer: 

Air 0°   20°        40°         60°         80°         100°   200° 

Mercury 0°    19°.98   39°.67    59°.62   79°.78    100°   202°.78 

Mercury  is  also  a  good  conductor  of  heat  and  has  a  low  specific  heat, 
causing  it  to  respond  rapidly  to  changes  of  temperature.  It  is  readily 
obtained  pure,  does  not  stick  to  glass,  and  remains  liquid  between  — 40° 
and  350°  C.  It  is  therefore  commonly  used  for  making  thermometers. 
Ether  and  alcohol  are  sometimes  used  for  sensitive  thermometers  on 
account  of  their  great  expansibility. 

As  time  goes  on  it  is  observed  that  there  is  a  gradual  change  in  the 
zero  of  a  thermometer,  caused  by  contraction  of  the  glass.  After  heating 
there  is  a  depression  of  the  zero,  since  the  glass  does  not  for  some  time 
resume  its  original  volume.  A  special  kind  of  glass  made  in  Jena  is  less 
subject  to  this  defect  than  other  kinds. 

Other  methods  of  measuring  very  high  temperatures  or  very  small 
differences  of  temperature,  depending  on  other  effects  of  heat,  will  be 
explained  later. 

CALORIMETRY. 

128.  Quantity  of  heat  may  be  relatively  measured  in  terms  of  any 
of  its  effects.      Being  a  form  of  energy,   the  most  scientific  method  would 
be  to  measure  it  in  terms  of  energy — in  ergs.     This  would  be  inconvenient 
in  practice,   however ;  hence  it  is  usually  measured  in  terms  of  physical 
effects  easily  produced.     Two  methods  of  calorimetry,   or  heat  measure- 
ment, are  ordinarily  used,  based  upon — 

1.  Change  of  State. — It  is    found  that  a  certain  amount  of  heat  is 
necessary  to  melt  a  given  mass  of  ice  or  other  substance,   or  to  vaporize 
it,   without  change  of  temperature.     It  is  expended  in  work  done  against 
molecular  forces.     It  may  be  assumed  that  to  change  the  state  of  twice  or 
n  times  the  mass  would  require  twice  or  n  times  as  much  heat.      By  this 
system  quantity  of  heat  would  be  actually  measured  in   terms  of  work 
done,  although  not  in  the  ordinary  work  units. 

2.  Change  of  Temperature. — If  a  certain  amount  of  heat  is  required 
to  raise  the  temperature  of  a  given  mass  of  a  substance  from  0°  to  1°,   it 
will  evidently  require  n  times  as  much  heat  to  raise  the  temperature  of  n 
times  the  mass  through  the  same  interval.      It  is  not  in  general  true,  how- 
ever, that  n  times  the  heat  will  raise  the  temperature  of  the  original  mass 
n  degrees,  although  it  is  approximately  true  in  most  cases. 

The  practical  unit  of  heat  is  the  calorie,  or  the  amount  required  to 
raise  the  temperature  of  one  gram  of  water  at  its  maximum  density 
from  4°  to  5°. 

The  thermal  capacity  of  a  body  is  the  amount  of  heat  required  to  raise 
the  temperature  of  a  body  one  degree. 

129.  The  specific  heat  of  a  substance  is  the  ratio  of  the  amount 
of  heat    required    to    raise    the    temperature   of  a    gram    one  degree  to 
that  required  to  raise  the  temperature  of  a  gram  of  water  one  degree, 


HEAT. 


or,  practically,  the  number  of  calories  required  to  raise  the  temperature  of 
one  gram  one  degree.  As  indicated  above,  this  quantity  is  usually 
variable  with  the  temperature,  although  for  practical  purposes  we  may 
assume  that  the  amount  of  heat  which  will  raise  the  temperature  of  n 
grams  of  water  from  4°  to  5°  will  raise  the  temperature  of  one  gram 
n  degrees.  The  variations  of  the  specific  heat  of  a  substance  are  due  to 
the  fact  that  not  all  of  the  heat  goes  into  increased  molecular  kinetic 
energy  (temperature),  a  part  being  expended  in  overcoming  molecular 
and  external  forces.  Only  in  the  case  of  gases  under  uniform  conditions 
of  pressure  or  of  volume  is  the  specific  heat  constant.  Black,  in  1760, 
discovered  the  fact,  previously  unsuspected,  that  the  specific  heat  of 
all  substances  is  not  the  same.  For  example,  if  100  grams  of  mercury 
at  100°  are  mixed  with  100  grams  of  water  at  0°  the  resulting  tempera- 
ture is  not  50°,  but  about  3°.  2.  In  general,  if  m  be  the  mass  and  s  the 
specific  heat  of  the  mercury,  M  the  mass  of  the  water,  and  t  the  final 
temperature, 

(102)  ms 


There  are  a  number  of  methods  of  determining  specific  heat. 

130.  Black'  s  ice  calorimeter  consists  of  a  block  of  ice,  with  an 
excavation  from  which  all  water  has  been  sponged  out.  Into  this  is 
placed  a  mass  m  of  a  substance  at  a  temperature  of  t  which  is  covered 
with  a  slab  of  ice.  In  falling  to  0°  the  heat  evolved  melts  a  mass  M  of 
ice,  which  can  be  removed  by  a  dry  sponge  and  weighed.  If  L  units  be 
required  to  melt  one  gram  of  ice, 


(103)  mst  =  LM=SQM. 

131.  The  method  of  mixture  is  most  commonly  employed.     If  m 
grams   of  a   substance   at   temperature    T  be   mixed    with  M  grams  of 
water  at  /0°  in  a  calorimeter  (thin  metal  vessel)  of  mass  mt  and  specific 
heat  slt  the  final  temperature  being  t  — 

(104)  ms  (T-^  =  (M+mIs2~)  (t-Q, 

from  which  s  may  be  calculated.  The  product  m^sx  is  called  the  water 
equivalent  of  the  calorimeter.  There  is  some  loss  by  radiation,  which 
may  be  corrected  by  observing  the  rate  of  cooling  of  the  water  at  its 
mean  temperature;  or  observations  of  temperature  may  be  made  at  regular 
intervals  while  stirring,  and  temperatures  and  times  plotted  as  coordinates. 
The  resulting  curve  beyond  its  maximum  ordinate  gives  the  law  of  cooling, 
and  if  continued  backward  until  it  cuts  the  axis  of  temperatures  it  will 
give  approximately  the  temperature  which  would  have  been  attained  had 
the  heat  been  imparted  instantaneously  and  without  loss  to  the  water. 
Radiation  losses  may  also  be  eliminated  by  making  the  initial  temperature 
of  the  water  about  as  much  below  the  temperature  of  the  room  as  its 
final  temperature  will  be  above  it.  This  is  easily  done  if  the  specific  heat 
of  the  substance  is  approximately  known  from  a  previous  experiment. 

132.  The   Bunsen   calorimeter  consists  of  a   bulb  of  glass   with  a 
laterally  communicating  capillary   tube,   and   a  test   tube  sealed  into  it. 
The  bottom  of  the  bulb  and  a  part  of  the  capillary  tube  are  filled  with 
mercury,  and  the  rest  of  the  bulb  with  water  from  which  all  air  has  been 
removed  by  boiling.     The  water  is  partly  frozen  around  the  test-tube  by 


A    Q  -    - 


-w*»^»^ 


«.  v*f 


-  C-v   '- 


SPECIFIC    HEAT.   V      ~  69 

X^t/FOfr 

placing  the  bulb  in  a  freezing  mixture.  If  m  grams  of  water  at  tempera- 
ture t°  be  placed  in  the  test-tube,  it  will  be  cooled  down  to  zero,  the  heat 
melting  the  ice  without  any  appreciable  loss  by  radiation.  The  volume  of 
the  contents  of  the  bulb  will  be  diminished,  and  the  mercury  column  in 
the  tube  will  recede  n  divisions.  Evidently  a  change  of  volume  of  one 
division  corresponds  to  the  absorption  of  mt  /  n  calories,  and  from  the 
amount  of  melting  produced  by  any  substance  introduced  into  the  test- 
tube  its  specific  heat  may  be  determined.  This  is  one  of  the  best  methods, 
especially  when  only  small  quantities  of  a  substance  are  available. 

133.  Joly  Steam   Calorimeter. — One  pan  of  a  balance  is  suspended 
in  a  steam-tight  chest  at  temperature  t°;  m  grams  of  the  substance  are 
placed  on  the  pan  and  counterbalanced;  dry  steam  is  admitted,  and  M 
grams  of  steam  are  condensed  on  the  pan  and  the  substance,  the  latent 
heat  set  free  raising  its  temperature  to  100°;  or  the  substance  may  be 
freely  suspended  from  the  balance-arm,  no  pan  being  used.     If  a  pan  is 
used  and  has  the  water  equivalent  m^s^  and  if  one  gram  of  steam  in  con- 
densing gives  out  536  calories — 

(105)  (ms  +  m,s^  (100  —  f)  =  LM=  536  M. 

134.  Specific   Heat  of  Gases. — When   a  compressed   gas   expands 
against  an  external  pressure  (no  heat  being  applied)  it  does  mechanical 
work,  at  the  expense  of  its  own  potential  energy.     If  we  place  a  thermo- 
pile or  sensitive  thermometer  in  such  an  expanding  gas  we  find  that  it  is 
cooled,  as  we  might  expect — for  if  there  is  no  other  source  of  energy,  the 
work  must  be  done  at  the  expense  of  the  kinetic  energy  of  the  molecules. 
If  the  compressed  air  in  a  vessel  saturated  with  water-vapor  be  allowed  to 
suddenly  expand,  the  water-vapor  will  be  condensed  in  a  cloud  by  the 
cooling.     Conversely,  if  a  gas  is  compressed  it  becomes  heated.     Rapid 
compression  of  air  containing  an  inflammable  substance  such  as  carbon- 
bisulphide  vapor  will  ignite  the  latter.     If  the  gas  expands  against  pressure, 
thus  doing  work,  we  must  supply  heat  to  maintain  the  temperature;  and 
if  we  wish  to  elevate  the  temperature  by  a  given  amount  it  will  take  more 
heat  than  if  the  gas  were  kept  at  constant  volume.     A  difference  should, 
therefore,   be   expected   between  the  specific  heat  of  a   gas   at   constant 
volume  and  that  at  constant  pressure.     This  is  found  to  be  the  case;  in 
fact,  there  are  an  infinite  number  of  specific  heats  of  a  gas,  depending  on 
the  conditions  of  pressure  and  volume.      If  rp  and  cv  be  the  two  principal 
specific  heats,  the  difference  between  them  will  be  the  value  in  heat  units 
of  the  work  done  on  unit  mass  in  expanding  against  the  constant  pressure 
p  while  the  temperature  rises  1°. 

In  general,  considering  m  grams  of  gas  and  calling  the  mechanical 
( ' '  Joule's  ' ' )  equivalent  of  heat  /,  if  the  gas  expands  a  distance  d  in  a  tube 
of  section  a,  against  a  pressure/, — 


(106)  „  (.,-..)  (T,-  T^=Work=-=  •. 
z^t  =  mR  T^  pvQ  =  mR  T0,  which  gives — 

(107)  *~*~T 

The  difference  between  the  two  specific  "heats  in  any  gas  is  equal  to  the 
gas  constant  R  divided  by  the  mechanical  equivalent  of  heat  (4.19  X  107 

ergs). 


70  HEAT. 

If  a  gas  expands  into  a  vacuum,  no  energy  is  expended  in  external 
work,  and  if  a  change  of  temperature  follows  it  means  that  molecular 
attractions  or  constraints  are  overcome  (if  temperature  falls),  or  that 
potential  energy  of  repulsion  has  been  transformed  into  kinetic  energy  (if 
the  temperature  rises).  In  a  "perfect"  gas  following  Boyle's  law,  there 
should  be  no  change  of  temperature. 

Joule  and  Thomson  found  that  in  all  the  gases  investigated  by  them, 
except  hydrogen,  there  was  a  very  slight  cooling,  indicating  a  feeble 
molecular  attraction.  W 

Regnault  determined  the  specific  heats  of  gases  at  constant  pressure  by 
forcing  them  from  a  large  reservoir  in  a  water  bath  through  a  spiral  tube 
immersed  in  a  calorimeter.  The  mass  of  gas  passing  in  a  given  time  can 
be  determined,  and  from  the  rise  of  temperature  in  the  calorimeter  the 
specific  heat  calculated.  The  specific  heat  at  constant  volume  may  be 
calculated  from  the  above  relation,  or  it  may  be  measured  directly  by 
Jolly's  steam  calorimeter.  A  considerable  mass  of  the  gas  is  forced  by 
pressure  into  a  copper  globe,  which  is  suspended  in  the  steam  chest.  To 
obviate  doubtful  corrections,  an  exactly  similar  but  exhausted  globe  is 
suspended  from  the  other  balance-arm  and  exposed  to  the  steam. 


Air 0.23741  0.1741 

Oxygen 21751  .1544 

Nitrogen  "      .24380  .1735 

Hydrogen 3.4090  2.4263 

It  will  be  observed  that  the  ratio  of  the  two  specific  heats  is  practically 
constant  for  all  gases  and  equal  to  1.4.  This  relation  will  be  explained 
later  (see  section  185).  The  difference  of  the  two  specific  heats  will  also 
be  found  to  correspond  closely  to  equation  (107). 

135.  Change  of  Specific  Heat  with  Temperature. — In    most   cases 
there  is  an  appreciable  change  in  specific  heat  with  temperature,  generally 
an  increase,  although  in  the  case  of  water  there  is  a  minimum  value  at 
about  30°.     The  specific  heat  of  diamond  is  about  three  times  as  great  at 
200°  as  it  is  at  0°.     Marked  changes  accompany  change  of  state,  as  shown 
in  the  case  of  water.      It  must  not  be  inferred  that  the  ' '  total  heat "  in  a 
body  is  equal  to  its  thermal  capacity  multiplied  by  its  absolute  tempera- 
ture.     Owing  not  only  to  variation  of  the  capacity  with  temperature,  but 
to  possibility  of  transformation  into  other  internal  forms  of  energy,  the 
expression  would  be  as  meaningless  as  ' '  the  total  amount  of  sound  in  a 
horn." 

The  high  specific  heat  of  water  causes  it  to  change  in  temperature 
very  slowly.  This  fact  has  a  great  influence  on  climate,  regions  on  the 
ocean  being  more  equable  than  those  inland — especially  if  they  are 
adjacent  to  warm  ocean  currents,  from  which  the  prevailing  winds  blow. 

136.  Latent  Heat. — The  amount  of  heat  absorbed  or  evolved  with- 
out change  of  temperature  when  a  unit  mass  of  a  substance  changes  its 
state  by  fusion,  vaporization,  solution,  etc.,  is  called  latent  heat,  although 
it   is  not  latent,    but  expended  or  given  out  by   work  done  upon  or  by 
molecular  forces.     The  latent  heat  of  a  substance  is  usually  determined  by 
the  method  of  mixtures.     If  m  grams  of  ice  be  melted  in  water  in  a  calorim- 


(1)  See  Ames,  The  Free  Expansion  of  Gases. 


MECHANICAL    EQUIVALENT    OF    HEAT.  71 

eter  whose  total  thermal  capacity  is  M  and  its  initial  and  final  tempera- 
tures T  and  /  — 

m  (L  +  t) 


If  m  grams  of  steam  be  condensed  in  the  calorimeter, 
(108)  m  (L  +  100  -  T)  =  M(  T-  f). 

The  latent  heat  of  ice  is  80  calories  ;  that  of  steam,  536.5. 

The  Bunsen  ice  calorimeter  may  be  used  to  determine  the  latent  heat 
of  ice,  knowing  the  densities  d^  and  d0  of  ice  and  of  water  at  0°.  If  Q 
calories  of  heat  be  supplied, 


(109) 


Q(\        1  \  ,       , 

-7-1-7  —  —  I  =v,  the  change  in  volume. 


137.  Mechanical  Equivalent  of  Heat.  —  In  absolute  scientific 
measurements  quantities  of  heat  may  be  expressed  in  foot-pounds,  kilogram- 
meters,  or  ergs.  In  the  first  case  the  practical  heat  unit  used  in  com- 
parison is  the  amount  which  will  raise  the  temperature  of  one  pound  of 
water  1°  Fahrenheit  ;  in  the  second,  the  amount  which  will  raise  the 
temperature  of  one  kilogram  1°  centigrade  (sometimes  called  the  greater 
calorie),  and  in  the  latter  case  the  calorie.  The  first  attempt  to  discover 
a  definite  relation  between  heat  and  work  was  made  by  Rumford,  from  his 
experiments  on  boring  cannon.  His  result  was  much  too  high,  being  847 
foot-pounds,  or  559.4  kilogram-meters.  Dr.  Robert  Mayer,  of  Heil- 
bronn,  in  1842  used  the  relation  cv  —  cv=p  (v^  —  v^lj  (equation  107), 
substituting  the  values  for  the  specific  heats  determined  by  Regnault. 
His  result  was  367  k  /  m.  The  validity  of  this  result  has  been  questioned 
by  English  physicists  on  the  ground  that  he  had  no  right  to  assume  that 
no  heat  was  expended  in  internal  work.  Joule  had  not  then  proved  this, 
but  it  seems  that  Gay-Lussac  had  performed  a  similar  experiment,  the 
results  of  which  were  known  to  Mayer. 

Joule  about  1842  began  a  classic  series  of  experiments  for  deter- 
mining the  mechanical  equivalent.  The  principal  method  used  by  him 
was  to  cause  a  falling  weight  to  rotate  a  paddle  in  a  calorimeter  filled  with 
water  or  mercury.  From  the  number  of  foot-pounds  of  work  done  and 
the  rise  of  temperature  of  the  calorimeter  the  ratio  7™  Wl  ff  could  be 
determined.  The  value  of  a  foot-pound  or  kilogram  -meter  depends  on 
the  local  value  of  g.  The  mean  result  of  his  work  reduced  to  the 
latitude  of  Greenwich  was  772.5  foot-pounds  or  424  k  j  m.  The  same 
method  on  a  more  elaborate  scale  was  tried  by  Professor  Rowland,  of 
Johns  Hopkins  University  in  1880.  The  paddle-wheel  was  driven  by  a 
steam-engine  and  made  n  revolutions  per  second.  The  work  done  was 
measured  by  a  torsion  wire,  a  weight  w  being  placed  at  a  distance  r  from 
the  axis  to  balance  the  force  of  friction.  The  work  done  was  2  tnwr,  and 
if  C  was  the  thermal  capacity  of  the  calorimeter  and  its  contents, 


(110)     /  =  =    **nwr    =  427.3>E>=-4.188  X  107  ergs  per  calorie. 

heat       L(/,  —  r0) 

Joule's  value  corrected  for  Baltimore  gives  7~  426.  75  /•  /;/. 
For  Berkeley  the  value  4.  19  X  107  ergs  per  calorie  may  be  used.  - 
Joule  and  others  have  also  determined  the  mechanical  equivalent  for 
the  heat  developed  by  a  given  quantity  of  electrical  energy  in  overcoming 


r 

HEAT. 

the  resistance  of  a  wire.     The  best  determination  by  this  method  was  made 
by  Griffiths,  with  a  result — 

(111)  /=427.45  kjm  for  Greenwich  =  4.193  x  107  ergs. 

Other  less  satisfactory  methods  have  also  been  employed,  such  as  the 
measurement  of  the  heat  developed  by  friction  or  percussion  of  metals, 
work  done  in  expansion,  etc. 

PROBLEMS. 

66.  How  many  ergs  are  there  in  a  foot-pound  ? 

67.  One  hundred  grams  of  lead  at  99°  are  dropped  into  100  grams  of  water 
at  15°  in  a  copper  calorimeter  weighing  40  grams.     What  is  the  final  temperature 
of  the  mixture  ? 

68.  A  mass  of  lead  falls  100  meters  on  a  plate  of  steel.     How  much  will  the 
temperature  of  the  lead  rise  ? 

69.  Calculate  the  value  of  J  from  the  given  specific  heats  of  hydrogen. 

70 .  Ten  grams  of  compressed  air  expands  against  a  pressure  of  5  atmospheres 
until  its  volume  is  increased  by  lOOc.c.     How  much  is  it  cooled  ? 

The  effects  of  heat  other  than  change  of  temperature  will  now  be  taken 
up  in  detail. 

138.  Change  of  Yolume  of  Solids. — Change  of  temperature  is 
usually  attended  by  change  of  volume — generally  dilatation  with  increase 
of  temperature. 

Measurement  of  Expansion. — Except  in  the  case  of  gases,  the  change  in 
dimensions  is  usually  small  and  requires  special  methods  of  measurement. 
In  measurements  of  lengths  there  are  two  principal  methods — that  of 
Lavoisier  and  Laplace,  in  which  one  end  of  a  bar  of  the  substance  is  kept 
in  a  fixed  position  and  the  other  end  in  contact  with  a  magnifying  lever. 
The  displacement  of  the  lever  may  be  directly  observed  or  determined 
from  the  angular  displacement  of  an  attached  telescope  and  the  elongation 
of  the  bar  computed;  Roy  and  Ramsderi s  method,  in  which  the  elon- 
gation is  directly  measured  by  a  micrometer  microscope. 

Coefficient  of  Expansion. — If  the  temperature  of  a  bar  is  raised  to  /°, 
its  initial  and  final  lengths  being  LQ  and  L  and  its  average  increase  in 
length  per  degree  / — 

L-L0  =  U 

L  =  L0  =  (l+T-t)=L.(l  +  af), 

in  which  a  is  called  the  mean  coefficient  of  linear  expansion  between  the 
limiting  temperatures,  or  the  average  increase  of  unit  length  for  one  degree. 
Isotropic  (non-crystalline)  bodies  expand  uniformly  in  all  directions. 
If  we  consider  a  rectangular  surface — 

S—  S0  =  L  (1  +  at)  £t  (1  +  O  —  LLl  =  S0  (2  at  +  a*t*}. 
When  a  is  very  small  the  last  term  may  be  neglected,  and 

(113)  5-  S0  (1  +  2  of)  =  S0  (1  +  6f). 

where  b  is  the  mean  coefficient  of  superficial  expansion. 

Similarly,  the  mean  coefficient  of  cubical  expansion  is  c  =  3  a. 

(114)  V=  F0 


CHANGE  OF  VOLUME  OF  SOLIDS.  73 

In  general,  these  coefficients  are  functions  of  the  temperature,  although 
the  change  in  value  within  ordinary  ranges  is  very  small. 

Below  are  given  the  linear  coefficients  of  expansion  of  some  substances. 
The  superficial  and  cubical  coefficients  are  obtained  from  these  by  multi- 
plying by  2  and  by  3  : 

Glass,    0°-100°,  0.0000086  Brass  at  40°,  0.0000186 

0°-200°,  92  Diamond,  5i2 

0°-300°,  101  Ebonite,  770 

Platinum  at  40°,  89  Paraffin,  2785 

Iron  122 

Glass  and  platinum  have  practically  the  same  coefficient,  so  that  when 
wires  have  to  be  sealed  in  glass  that  metal  is  generally  used. 

As  a  rule,  coefficients  of  expansion  increase  with  temperature  as 
measured  by  the  gas  themometer.  A  general  expression  for  the  volume 
at  a  given  temperature  would  be  — 

(115)  V=  V0(l  +  At+Bt*+  03+-   •   •), 

in  which  A,  B,  and  C  are  small  constants  to  be  determined  from  experiment. 

Great  force  is  exerted  by  thermal  changes  of  length  of  volume.  A  rod 
contracting  a  given  length  when  cooled  exerts  the  same  force  that  would 
be  required  to  stretch  it  the  same  amount,  as  determined  by  Young's 
modulus. 

The  coefficient  of  expansion  of  a  given  substance  is  not  always  the 
same  for  different  specimens;  it  depends  somewhat  on  temper,  im- 
purity, etc. 

Owing  to  their  different  rates  of  expansion,  bars  of  different  materials 
riveted  together  will  bend  one  way  or  another  with  change  of  tempera- 
ture. Breguet's  thermometer  is  such  a  compound  ribbon  in  spiral  form, 
which  twists  or  untwists  with  change  of  temperature. 

In  Harrison's  compensated  gridiron  pendulum  the  length  of  the 
pendulum  is  constant  if  Z,I?  L2,  and  L3  conform  to  the  condition, 


since  these  quantities  represent  changes  of  length  in  opposite  directions;  or 


139.  Anomalous  Expansion.  —  Rubber  under  tension  contracts  in  the 
direction  of  tension  when  heated,  but  it  expands  more  at  right  angles;  so  that 
on  the  whole  there  is  an  increase  of  volume.  Iodide  of  silver  is  found  to 
contract  when  heated  between  —  10°  and  70°,  although  chloride  and 
bromide  of  silver  expand  normally.  In  the  case  of  the  iodide  some  of  the 
coefficients  in  equation  (115)  are  negative,  so  that  the  expression  changes 
sign  about  —  60°,  showing  that  below  that  temperature  expansion  is  normal. 
Similar  equations  indicate  that  diamond  and  emerald  have  maximum 
densities  at  —  41°.  7  and  —  4°.  2  respectively;  below  those  temperatures 
they  may  expand  when  cooled;  but  we  must  be  careful  in  drawing 
inferences  beyond  the  range  of  experiment.  Such  anomalous  cases  are, 
no  doubt,  the  result  of  crystalline  structure. 

Alloys  often  show  anomalous  expansion.  Nickel-steel  (36$  nickel) 
has  a  coefficient  of  only  0.000001,  which  makes  it  useful  for  measuring 
instruments. 


74  HEAT. 

Expansion  of  Crystals. — In  general,  the  coefficients  of  expansion  of 
crystals  differ  along  different  axes,  being  the  same  in  all  directions  only  in 
the  isometric  system ;  consequently  a  sphere  cut  from  a  crystal  will  in 
general  assume  an  ellipsoidal  shape  with  change  of  temperature.  In  some 
cases  the  coefficient  along  one  axis  may  be  negative ;  for  example  : 

Beryl  Iceland  spar 


Along  principal  axis  +  0.000001722  +  0.0000263 

Perpendicular  to  principal  axis  -  0.000000134  —  0.0000031 

In  every  case  (with  the  exception  of  iodide  of  silver)  there  is  increase 
of  volume  of  crystals  with  rise  of  temperature. 

140.  Expansion  of  Liquids.  —  As  a  rule,  the  rate  of  expansion  of 
liquids  is  greater  than  that  of  solids.  They  must  be  contained  in  solid 
vessels,  so  that  allowance  must  be  made  for  the  expansion  of  the  vessel  in 
determining  the  coefficient.  This  is  avoided  in  the  method  of  balanced 
columns,  used  by  Dulong  and  Petit  and  by  Regnault.  One  form  of 
apparatus  used  by  the  latter  consists  essentially  of  a  pair  of  U-tubes,  the 
short  arms  of  which  are  in  communication  with  an  air  chamber  in  which 
any  desired  pressure  may  be  maintained,  while  a  small  horizontal  tube 
connecting  the  longer  arms  keeps  the  mercury  in  them  at  the  same  level. 
The  two  tubes  are  placed  in  separate  baths  of  water  or  other  liquid  which 
may  be  kept  at  any  desired  temperatures.  When  in  equilibrium, 

h,P,  =  h2P2=p  /g. 

But  P.Cl  +  'O^a+'O^o- 

Therefore,   £r(l  +  cQ  =  /i2(l  +  rfx)  . 


Regnault  found  for  the  mean  coefficient  of  dilatation  of  mercury 
between  0°  and  f  — 

c  =  0.0001791  +  0.000000025/, 

which  is  practically  constant  between  0°  and  100°. 

The  Weight  Thermometer.  —  A  glass  bulb  is  filled  with  M  grams  of  a 
liquid  at  0°.  The  bulb  is  then  placed  in  a  steam  or  water  bath  at  temper- 
ature t°  and  m  grams  of  the  liquid  are  driven  out  by  expansion.  Assuming 
no  change  in  the  volume  of  the  bulb  and  calling  the  apparent  coefficient  of 
expansion  a  — 

F0  =  M/PO  =  (M-  ni)/P-=  (M~  nt}  (1  +  at}. 

^o 

Therefore,   a  = 


(M—  m)  t 

But  if  the  glass  has  a  coefficient  g,  and  the  true  coefficient  of  expan- 
sion of  the  liquid  is  c  — 

(118)  V-  VQ  (1  +  cf)  -=^I^L  (1  +  af)  (!+£•/) 


and  c  =  a  +£*.  approximately. 

Weighing.  —  Hallstrom  and  Matthiessen  determined  the  density  of 
liquids  by  weighing  in  them  at  different  temperatures  a  cube  of  glass  of 
known  coefficient.  The  loss  of  weight  of  the  glass  being  equal  to  the 


EXPANSION    OF    LIQUIDS — EXPANSION    OF   GASES.  75 

weight  of  an  equal  volume  of  the  liquid,  its  density  can  be  determined  at 
any  temperature,  and  the  coefficient  of  expansion  computed. 

The  expansion  of  water  is  anomalous.  It  contracts  when  cooled  down 
to  about  4°  C. ;  below  that  point  it  expands  until  it  freezes,  after  which  it 
contracts  normally  with  reduction  of  temperature.  The  density  of  water 
at  various  temperatures  was  determined  by  Hallstrom  by  the  method  of 
weighing  a  piece  of  glass  in  it.  He  found  the  following  to  express  the 
density  at  any  temperature  /  in  terms  of  the  density  at  zero : 

(119)      P  =  PO(!  +0.000052939*—  0.00000653/2  +  0.00000001445/3), 

which  gives  a  maximum  density  at  4°.  118. 

Despretz  determined  the  density  directly  by  a  water  thermometer.  A 
curve  is  drawn  with  apparent  volumes  and  temperatures  as  coordinates. 
Another  curve  is  drawn  showing  the  change  in  volume  of  the  glass.  A 
third  curve,  whose  ordinates  are  the  sum  of  those  of  the  first  two,  gives 
the  true  expansion  of  water,  and  a  minimum  is  found  at  4°. 007.  In 
another  experiment  Despretz  used  four  thermometers,  placed  at  different 
depths  in  a  vessel  of  water,  which  was  then  cooled.  Curves  were  drawn 
for  each  thermometer,  with  temperatures  and  times  as  coordinates.  The 
lower  thermometer  came  to  a  stationary  condition  when  the  water  around 
it  came  to  the  maximum  density ;  likewise  with  the  others,  until  the  tem- 
perature of  the  top  thermometer  reached  the  same  point,  when  all  began 
to  fall.  The  common  intersection  of  the  curves  shows  the  temperature  of 
maximum  density,  which  was  found  to  be  3°.  974. 

The  expansion  of  water  in  freezing  plays  an  important  part  in  disin- 
tegrating rocks  and  soil.  The  fact  that  the  temperature  of  maximum 
density  is  above  the  freezing  point  prevents  bodies  of  water  from  freezing 
solid.  It  has  been  suggested  that  the  expansion  of  water  below  4°  is  due 
to  the  gradual  formation  of  ice  crystals  at  that  temperature.  The 
readjustment  of  the  molecules  in  the  crystalline  configuration  requires 
greater  space.  Increase  of  pressure  (which  would  naturally  retard  the 
formation  of  these  crystals)  lowers  the  temperature  of  maximum  density. 
For  93  atmospheres  the  temperature  is  2°;  at  145  atmospheres,  0°.6. 

In  saline  solutions  the  point  of  maximum  density  (as  well  as  the 
freezing-point)  is  lowered  proportionally  to  the  amount  of  salt  in  solution. 
Some  results  for  sodium  chloride  solution  are  given  below : 

Percentage  of  salt         Maximum  density  Freezing-point 

0  4°  0° 

1  10.77  — 0°.65 
4  —5°.  63  —2°.  60 
8  —16°.  62  —5°.  12 

The  behavior  of  bismuth  is  similar  to  that  of  water. 

141.  Expansion  of  Oases. — It  has  been  shown  that  all  gases  at 
constant  temperature  very  closely  conform  to  Boyle's  law:  pv  =  constant. 
It  has  also  been  demonstrated  that  the  pressure  of  a  gas  may  be  explained 
as  the  result  of  molecular  impact,  and  that  it  is  directly  proportional  to 
molecular  kinetic  energy.  If  either  the  pressure  or  the  volume  of  a  gas 
be  kept  constant,  the  other  factor  in  the  above  expression  varies  in  the 
same  arithmetical  progression ;  therefore  the  volume  of  a  gas,  if  the 
pressure  be  kept  constant  varies  as  the  molecular  kinetic  energy.  Experi- 
ment also  shows  that  changes  in  the  kinetic  energy  of  a  gas  (measured 


76  HEAT. 

by  pressure)  are  proportional  to  the  increments  of  heat.  Joule  and 
Thomson  proved  that  when  a  gas  expands  into  a  vacuum  (thus  doing  no 
external  work)  its  temperature  is  very  slightly  changed,  showing  thai 
there  is  practically  no  internal  molecular  work.  Nevertheless,  we  can 
not  say  that  the  increase  in  molecular  kinetic  energy  is  equal  to  the 
amount  of  heat  absorbed,  for  the  reason  that  some  work  may  be  done 
inside  the  molecules,  on  the  atoms.  Since,  then,  the  changes  in  volume 
of  a  gas  at  constant  pressure  are  proportional  to  the  amount  of  heat  added, 
a  gas  thermometer  has  a  decided  advantage  over  one  of  any  other  sub- 
stance, in  which  changes  in  volume  are  not  proportional  to  the  absorption 
of  heat.  Moreover,  all  gases  have  practically  the  same  rate  of  expansion, 
and  we  can,  if  we  choose,  use  variations  in  pressure  at  constant  volume  to 
determine  temperatures. 

The  statement  of  Boyle's  and  Charles'  laws  combined  is  : 


If  p  be  kept  constant, 

(120)  a  =—    —  —  (coefficient  of  expansion). 

^o^ 

If  v  be  kept  constant, 

(121)  a2  =        j    (coefficient  of  increase  of  pressure)  . 

Variations  from  the  gaseous  laws  are  indicated  by  differences  in  these 
coefficients.     In  a  perfect  gas  they  would  be  the  same.     Measurements  of 
the    coefficients    have    been    made   by    Gay-Lussac,    Dalton,    Magnus, 
Regnault,   Amagat,   and  others.     Some  of  Regnault's  results  are  given 
below  : 

ax  (atmospheric  p.).        a2  (in  neighborhood  of  at.  p.). 
Air  ..................     0.0036706  0.0036650 

Hydrogen  .........  36613  36678 

Carbon  dioxide...  37099  36896 

Sulphurous  oxide  3903  3845 

Regnault's  experiments  show  that  the  more  highly  rarefied  the  gas 
and  the  higher  the  temperature,  the  nearer  a^  and  a2  approach  the  same 
value  for  all  gases  —  that  is,  the  more  nearly  Boyle's  law  is  obeyed. 

142.  Absolute  Zero.  —  In  the  expression  pv=p0v0(\+  af)  the 
second  term  becomes  zero  at  a  temperature  t  =  —  1  /  a  centigrade.  This 
may  be  interpreted  to  mean  that  if  a  gas  strictly  obeyed  Boyle's  law  at  all 
temperatures,  the  pressure  of  a  finite  volume  of  the  gas  would  become 
zero.  It  is  impossible  to  conceive  of  a  lower  temperature  than  this, 
which  is  accordingly  called  the  absolute  zero  of  temperature.  Substi- 
tuting values  of  a  determined  by  experiment  at  different  temperatures  in 
the  above  expression,  Regnault  found  the  following  values  for  the  absolute 
zero:  Air  at  76  cm.  pressure,  t  =  —  272°.  85;  at  149  cm.  pressure, 
^  =  —  272°.  70.  Corrections  for  deviations  from  Boyle's  law  make  the 
value  of  /  about  273°.  Reckoning  temperatures  from  the  absolute  zero, 
we  have: 

(122)  pv  =  ap0v0(\  /  0  +  0  =  ap0v0  T=  mR  T, 

in  which  T  represents  temperature  on  the  absolute  scale. 


INTERNAL  WORK — FUSION  AND   SOLIDIFICATION.  77 

Gas  thermometers  are  of  two  kinds — constant  pressure,  in  which 
variations  of  volume,  and  constant  volume,  in  which  variations  of  pressure, 
are  proportional  to  the  absolute  temperature.  As  a  rule  gas  thermometers 
are  used  only  for  correcting  mercurial  thermometers,  which  are  directly 
employed  in  scientific  work. 

The  coefficient  of  expansion  of  liquids  near  the  point  of  vaporization 
may  exceed  that  in  the  gaseous  state.  For  example,  Thilorier  found  for 
carbonic  dioxide  between  0°  and  30°  C. — for  liquid,  #=.016;  for  gas,  .0037. 
Wroblewski  found  for  liquid  oxygen  at  — 139°,  a  =  .017;  liquid  nitrogen  at 
— 154°,  #  =  .0311,  these  values  greatly  exceeding  those  for  the  gases. 
Dewar  found  for  solid  hydrogen  a  coefficient  of  expansion  ten  times  that 
of  the  gas. 

QUESTIONS   AND   PROBLEMS. 

71.  Calculate  the  change  in  length  of  the  railroad  tracks  between  Berkeley 
and  Oakland  (say  5  miles)  caused  by  a  temperature  change  of  20°. 

72.  What  change  of  period  of  a  brass  seconds  pendulum  is  caused  by  a  change 
of  10°  in  temperature  ? 

73.  In  a  compound  pendulum  of  brass  and  iron  the  iron  grids  are  in  the  aggre- 
gate 100  cm.  long.     How  long  must  the  brass  grids  be  to  secure  compensation? 

74.  How  is  the  principle  of  expansion  utilized  in  putting  on  wagon  tires;  in 
fitting  together  the  concentric  steel  cylinders  of  modern  cannon;  in  heating  rivets 
before  hammering  them  in  ? 

75.  What  is  the  volume  of  a  gram  of  air  at  20°  and  under  100  cm.  pressure  ? 

76.  A  mass  of  aluminum  is  counterbalanced  by  a  brass  kilogram  weight  when 
the  temperature  is  20°  and  the  pressure  76  cm.     What  is  the  weight  of  the  alumi- 
num in  vacuum  ? 

CHANGE   OF   STATE. 

143.  Internal  Work. — In  the  case  of  gases  the  absorption  of  heat 
results  merely  in  increase  of  temperature  if  no  external  work  is  done,  no 
energy  being  expended  in  overcoming  molecular  forces.     In  the  case  of 
liquids  and  solids  heat  is  expended  not  only  in  increasing  kinetic  energy 
of  the  molecules,  but  in  overcoming  their  constraints.     In  general,  at  a 
certain  temperature  depending  on  the  substance  and  the  conditions,  a 
point  is  reached  beyond  which  no  increase  of  molecular  agitation  can  take 
place  without  disruption;  or,  if  the  substance  is  cooling,  the  diminished 
motion  of  the  molecules  allows  them  to  yield  to  their  mutual  attractions. 
At  this  point  a  change  of  state  takes  place,  heat  being  absorbed  or  given 
out  by  the  internal  work  done  upon  or  by  the  molecular  forces.     As  a 
rule,  during  a  change  of  state  there  is  no  change  of  temperature;  hence 
the  erroneous  name  ' '  latent ' '  applied  to  the  heat  transformed  to  work  or 
conversely.     The  name  was  first  used  by  Dr.  Black  about  1757,  on  the 
assumption  that  heat  is  a  material  fluid. 

144.  Kinds  of  Change. — Fusion  or  solution,  passage  from  solid 
to  liquid  state;  converse  operations,  freezing  and  crystallization.     Vapor- 
ization, liquid  to  gas,  or  sublimation,  solid  to  gas;  converse,  condensation. 
Other  less  definite  modifications  due  to  heat  are  molecular  dissociation 
and  ' '  allotropic ' '  changes  of  structure. 

145.  Fusion    and    Solidification. — All  known  elementary  sub- 
stances except  carbon  and  molybdenum  have  been  melted.     There  are 
two  classes  of  substances  with  regard  to  condition  of  melting.     The  first 
class,   comprising  all  substances  of  crystalline  structure,   have  a  definite 
melting-point,  the  temperature  remaining  constant  until  the  process  is 


78  HEAT. 

complete  (ice,  sulphur).  The  second  class  comprises  most  non-crystalline 
substances,  such  as  wax,  glass,  iron.  They  gradually  soften  on  the  appli- 
cation of  heat,  the  temperature  rising  continuously  until  fusion  is  com- 
plete. The  following  laws  apply  to  the  first  class: 

Laws  of  Fusion. — 1.  All  crystalline  substances  have  a  definite  melting 
(freezing)  point.  2.  Until  the  process  is  completed  the  temperature 
remains  constant.  3.  Each  unit  mass  of  the  substance  in  melting  (or 
freezing)  absorbs  (or  evolves)  a  definite  amount  of  heat,  called  its  latent 
heat  of  fusion. 

146.  Surfusion.      Under  certain  conditions  liquids  may  be  cooled 
below  their  normal  freezing-points  without  freezing.      Fahrenheit  cooled 
pure  water  in  a  bulb  to  — 13°;  it  froze  on  breaking  the  bulb.     Gay-Lussac 
reduced  the  temperature  of  water  beneath  a  layer  of  oil  to — 12°;  agita- 
tion caused  immediate  solidification.      Melted  phosphorus  may  be  cooled 
far  below  its  freezing-point,   but  immediately  solidifies  on  introducing  a 
particle  of  solid  phosphorus.     In  all  cases  of  surfusion,  agitation  or  the 
addition  of  a  particle  of  the  solid  causes  solidification.      Perhaps  small  ice 
crystals  are  already  in  suspension,  which  are  thus  brought  together. 

When  freezing  sets  in,  the  liberation  of  latent  heat  at  once  brings  the 
temperature  up  to  the  normal. 

Melting-point  Latent  h,fratjn  calories 

per  £>rcirrl. 

Hydrogen —258°                                             

Chlorine —102                                               

Mercury —39  2.8 

Water 0  80.0 

Phosphorus 44  4.7 

Sulphur !..  115  9.4 

Bismuth 264  12.4 

Lead 328  5.3 

Antimony 425                                                

Copper 1082                                             

Cast  iron 1000-12001                                  

Steel 1300-1400  I    Gradual                

Iron  (wrought) 1500-1800  [softening.               

Platinum 1800-2200  j                                  

Iridium  1950                                              

The  possibility  of  welding  depends  upon  plastic  condition  before 
fusion,  as  in  the  case  of  iron,  platinum,  and  glass. 

147.  Alloys. — Many  alloys,  like  amorphous  solids,  soften  gradually,, 
the  most  fusible  constituent  melting  first.      Others  have  a  fairly  definite 
melting-point,  usually  lower  than  that  of  any  of  their  constituents.      For 
example,  Wood's  fusible  metal — composed  of  bismuth,  4  parts;  lead,  2; 
cadmium  and  tin,  1  each— melts  at  about  60°.     Rose's  fusible  metal  melts 
at  94°.     Steel  and  cast  iron,   containing  carbon,   melt  at  a  much  lower 
temperature  than  wrought  iron.     Alloys  may  be  considered  as  solutions  of 
solids  in  solids.     In  all  solutions  there  is  a  lowering  of  the  freezing-point. 

148.  Change  of  Volume. — In  most  cases  fusion  is  attended  by  an 
increase  of  volume;  but  there  are  a  few  exceptions,  such  as  ice,  bismuth, 
antimony,   brass,   iron.      Metals  like  these  are  best  for  casting,   as  they 
give  sharp  impressions.      For  this  reason  bismuth  and  antimony  are  used 
in  type  metal.      In  general,   the  rate  of  expansion  of  the  liquid  is  greater 
than  that  of  the  solid. 


SOLUTION    AND    CRYSTALLIZATION.  79 

149.  Effect  of  Pressure. — A  change  of  pressure  in  general  affects 
the  freezing-point   slightly.     Water  expands   on   freezing ;  therefore   an 
increase  of  pressure  opposes  solidification  and  lowers  the  freezing-point. 
The  lowering  due  to  1  atmosphere  pressure  is  0°.0072.     Mousson  inclosed 
water  in  a  stout  steel  tube,   with  a  piece  of  copper  at  the  bottom.     The 
water  was  frozen  at  — 20°.     The  tube  was  inverted  and  a  pressure  of 
13 *  000  atmospheres  applied.     On  removing  the  pressure  and  opening  the 
tube  the  water  was  frozen;  but  the  piece  of  copper  was  at  the  end  opposite 
to  that  originally  occupied,   showing  that  the  ice  had  melted.      Pressure 
has  an  opposite  effect  on  substances  which  expand  when  melted.     In  such 
cases  increase  of  pressure  tends  to  maintain  the  solid  state,  and  the  melting- 
point  is  raised,      Paraffin  melts  at  46°  under  ordinary  conditions  and  at 
50°  under  100  atmospheres  pressure.     It  is  believed  that  the  rocks  in  the 
interior  of  the  earth  may  for  this  reason  remain  solid  even  at  temperatures 
far  above  their  normal  melting-point. 

150.  Regelation. — When  two  pieces  of  ice  are  pressed  together  they 
are  liquefied  at  the  bounding  surface ;  removing  the  pressure,   they  freeze 
together.      Faraday  called  this  regelation.     This  is  illustrated  by  snow- 
balls, which  can  be  squeezed  into  compact  masses  of  ice,  or  by  attaching 
weights  to  a  wire  loop  surrounding  a  block  of  ice.     The  ice  melts  under 
the  wire,  flows  around,  and  freezes,  so  that  when  the  wire  has  worked  its 
way  through,  the  block  is  still  solid.     The  flow  of  glaciers  is  partly  due 
to  the  same  cause.     At  any  point  where  pressure  exists,  such  as  at  the 
bottom  or  sides,   the  ice  is  melted,  flows  around  the  obstacle,  and  again 
freezes.     Their  motion  is  also  partly  due  to  expansion  downward  by  the 
sun's  heat,  followed  by  contraction  in  the  same  direction. 

151.  Solution  and  Crystallization. — The  liquefaction  of  a  salt 
by  solution  is  essentially  the  same  process  as  fusion,  and  is  accompanied 
by  an  absorption  or  evolution  of  latent  heat,   although  this  result  may  be 
masked  by  thermal  changes  due  to  chemical  action.     The  apparent  latent 
heat  may  thus  be  either  positive  or  negative,  but  the  true  latent  heat  is 
always  negative  (absorbed  on  solution). 

152.  Freezing-points   of  Solutions. — It  is  found  that  the  freezing- 
points  of  solutions  are  lowered  in  proportion  to  their  concentrations  (so 
long  as  these  are  small),  the  forces  between  dissimilar  molecules  prevent- 
ing the  formation  of  crystals.     Raoult  found  about  1882  that  in  most 
cases,   especially  in  solutions  of  organic  substances,   such  as  sugar,  the 
lowering  of  the  freezing-point  was  the  same  when  concentrations  of  the 
solutions  of  different  substances  were  made  proportional  to  their  molec- 
ular weights,  (equimolecular).     This  shows  that  all  molecules,  whatever 
their  nature,  have  the  same  effect  in  lowering  the  freezing-point.     This  is 
approximately  true  for  some  alloys.     The  divergences  from  this  law  in  the 
case  of  solutions  of  inorganic  salts  or  acids  have  been  explained  as  the 
result  of  partial  dissociation,  each  atom  having  the  same  effect  as  an  entire 
molecule.     These  deviations  are  of  the  same  magnitude  as  the  deviations 
from  normal  osmotic  pressure  ascribed  to  the  same  cause.     (See  section 
103.)     This  principle  is  useful  in  comparing  molecular  weights.  W     Gen- 
erally when  a  solution  is  frozen  the  ice  excludes  the  salt,  except  small  traces 
mechanically  suspended.     Ice  from  a  colored  solution  is  colorless. 

Freezing  Mixtures. — If  a  salt  be  mixed  with  broken  ice,   a  solution 
will  be  formed  owing  to  the  breaking  down  of  the  crystals  by  the  attrac- 

(1)     See  Jones,  Theory  of  Solutions;  Ostwald,  Solutions  ;  Whetham,  Solutions. 


VAPORIZATION    AND    CONDENSATION.  81 

that  point.  At  a  certain  temperature  the  vapor  pressure  will  equal  this 
sum,  and  from  that  time  there  will  be  formation  of  vapor  bubbles  through- 
out the  mass.  This  distinguishes  boiling  or  ebullition  from  evaporation. 
Latent  Heat.  When  a  liquid  begins  to  boil,  the  heat  which  is  supplied 
to  it  is  wholly  used  in  doing  work  against  molecular  forces  and  the 
temperature  of  the  liquid  remains  constant.  Of  course  some  heat  dis- 
appears in  this  way  in  evaporation  at  lower  temperatures. 

156.  Laws  of   Vaporization. — 1.   Liquids  evaporate  at  all  tempera- 
tures.     Every  liquid  begins  to  boil  when  the  temperature  is  such  that  the 
pressure  of  its  vapor  begins  to  exceed  the  external  pressure      2.   So  long 
as   the    pressure   remains    constant   the    temperature   remains   constant. 
3.   To  change  a  unit  mass  of  liquid  to  vapor  (or  conversely)  a  definite 
amount  of  heat  is  absorbed   (or  evolved) ,    called  latent  heat  of  vapori- 
zation. 

The  normal  boiling-point  of  a  liquid  corresponds  to  a  pressure  of  76 
cm.  of  mercury.     Below  are  some  values  of  the  normal  boiling-point. 

Hydrogen -258°  Water  100° 

Oxygen -181°.4  Mercury 357°.25 

CO2 -80°  Sulphur 448°.4 

SO2 -10°  Zinc 891°-1040° 

Ether  +35°  Lead  1500° 

Alcohol  78° 

157.  Distillation. — Mixed  substances  having  different  boiling-points 
may  be  separated  by  maintaining  the  temperature   of  the  mixture  just 
above  the  boiling-point  of  one  constituent  until  it  has  all  been  vaporized  ; 
then  raising  the  temperature  just  above  the  boiling-point  of  the  next  and 
so  on.     Some   of  the  substances  having  higher  boiling-points  pass  off  by 
ordinary  evaporation.     At  each  repetition,  however,  the  fraction  is  reduced ; 
hence  the  term  fractional    distillation.      In  similar   chemical   compounds 
there  is  often  a  definite  relation  between  boiling-points.     For  example,  in 
certain  homologous  carbon  compounds  there  is  a  rise  of  about  19°  for 
each  CH2  radical  added  to  the  molecule. 

158.  Superheating. — Distilled  water   free    from    air  may    have    its 
temperature  raised  above  the  boiling-point  without  ebullition.     Mechanical 
disturbance  or  the  introduction  of  impurities  cause  boiling  with  explosive 
violence,  and  the  temperature  then  drops  down  to  the  normal.     This  is 
illustrated   in   the   "bumping"  of  pure  water  in  a  clean  glass  beaker. 
The  introduction    of  sand  or  broken  glass  causes  normal  boiling.     Air 
bubbles  or  sharp  points  seem  to  act  as  nuclei  for  the  aggregation  of  vapor 
molecules  into  bubbles.     Drops  of  water  in  a  mixture  of  linseed  oil  and 
oil  of  cloves  have  been  raised  to  178°  without  boiling.     In  the  last  case 
surface    tension   plays  an   important  part.      In  addition  to   the  external 
pressure,  the  vapor  must  overcome  the  pressure  p  =  2  T j  r.     (Surface 
tension  promotes  surface  evaporation  from  a  convex  surface,  but  boiling 
takes  place  internally. —  See  section  166.)    The  steam  from  boiling  water 
always  has  the  same  temperature  under  the  same   pressure ;  hence   in 
calibrating  thermometers  they  are  suspended  in  steam. 

159.  Supersaturation. — Aitken  has  shown  that  if  a  vapor  be  entirely 
free  from  suspended  dust  particles  or  water  drops,   it  may   be  cooled 
considerably  below  its  ordinary  condensing  point  without  condensation. 
Such  particles  seem  to  act  as  nuclei  for  the  formation  of  drops.     Surface 
tension  plays  some  part  in  this.      (See  section  166.) 


82  HEAT. 

160.  Boiling-point  of  Solutions. — It  is  found  that  the  boiling-point 
of  solutions  is   raised   in   proportion   to  their  concentrations.      The  same 
attraction  between  dissimilar  molecules  which  promotes  solution  of  solids 
resists  their  separation  by  vaporization.      Raoult  found  the  same  law  to 
apply  as  in  the  case  of  lowering  of  the  freezing-point.     In  solutions  of 
different  salts  of  concentrations  proportional  to  their  molecular  weights 
the  raising  of  the  boiling-point  is  the  same,  or  each  molecule  has  the  same 
effect,   whatever  the  substance.     Since  only  pure-water  vapor  passes  off, 
with  a  pressure  corresponding  to  the  temperature,  we  see  that  for  a  given 
temperature  of  solution  there  is  a  lowering  of  the  vapor  pressure  propor- 
tional directly  to  the  concentration   (so  long  as  this  is  not  great)  and 
inversely  to  the   molecular  weight.     This   principle  is  used  in  the  com- 
parison of  molecular  weights.     It  applies  also  to  the  vapor  pressure  of 
solutions  of  some  metals  in  mercury  (amalgams) .  W 

161.  Spheroidal  State. —  A   small    quantity    of  liquid  placed    on  a 
surface  heated  considerably  above  its  boiling-point  does  not  immediately 
boil  away,   but  assumes  a  spheroidal  shape,  vibrating  and  rolling  on  the 
surface.     This  phenomenon  was  first  investigated  by  Leidenfrost  and  later 
by  Boutigny.      Close  observation  shows  that  the  drop  is  separated  from 
the  surface  by  a  cushion  of  vapor.     It  has  been  found  by  introducing  the 
bulb  of  a  small  thermometer  into  a  drop  that  the  temperature  of  the  lower 
part  did  not  exceed  98°  and  that  of  the  upper  about  90°,  showing  that  the 
conduct  of  the  substance  is  not  due  to  the  assumption  of  a  new  state,   as 
once  supposed.      If  liquid  sulphur  dioxide  be  placed  in  red-hot  crucible  it 
will   assume   the   spheroidal   state  at  a  temperature  below  the  freezing 
temperature  of  water.     If  a  small  quantity  of  the  latter  be  poured  above 
it,  the  water  will  be  frozen.     In  the  same  way  the  hand,  if  wet,   may  be 
plunged  into  melted  metal  without  injury,   being  protected  by  the  vapor. 

162.  Sublimation. — Ice  and  snow  in  the  arctic  regions  evaporate  with- 
out melting ;  in  this  case  the  temperature  is  too  low  for  the  liquid  to  exist. 
Under  ordinary  conditions  arsenic,   camphor  and  iodine  will  pass  directly 
into  vapor ;  by  increasing  the  pressure  they  may  be  made  to  pass  through 
the  liquid  state.     Carbon  will  vaporize  in  the  electric  arc,  but  has  never 
been  liquefied.     In  general,  sublimation  occurs  when  the  pressure  of  the 
vapor  at  the  freezing-point  is  greater  than  the  external  pressure.     Under 
such  conditions  the  liquid  cannot  exist.     Substances  which  sublime,  such 
as  salammoniac,  show  phenomena  resembling  those  of  the  spheroidal  state. 

163.  Latent  heat   of   vaporization   is    usually    determined    by    the 
method  of  mixture.     It  is  found  to  be  variable  with  the  temperature  at 
which  boiling  occurs.     For  water  Regnault  found  that  the  expression — 

(123)      Z,  =  606.5  —  0.695 1  —  0.00002 12  —  0.0000003 1* 

holds  up  to  t  —  230°.  This  expression  vanishes  at  706°,  indicating  a 
direct  passage  from  fluid  to  gaseous  state  without  expenditure  of  work 
(see  section  172).  Some  values  for  latent  heat  of  vaporization  under 
ordinary  conditions  are:  Water,  536;  alcohol,  202;  ether,  90.4. 

164:.  Cooling  Due  to  Evaporation. — When  evaporation  takes 
place  without  a  supply  of  heat  from  an  external  source,  cooling  of  the 
liquid  and  its  surroundings  takes  place.  Water  in  porous  earthenware 
vessels,  shallow  dishes,  or  canvas-covered  jars  (ollas)  is  cooler  than  if  kept 
inclosed.  Water  will  freeze  under  the  exhausted  receiver  of  an  air  purnp^ 

(' )  S_c  w  -rks  on  Solutions  by  Jones,  Ostwald  and  Whet>am. 


84  HEAT. 

of  radius  r,  and  shows  that  the  equilibrium  pressure  around  a  drop  is 
greater  than  for  a  plane  surface ;  hence  a  drop,  if  very  small,  will  evapo- 
rate even  into  saturated  space.  This  shows  why  condensation  is  aided  by 
dust  particles,  etc. ,  (the  moisture  spreading  over  the  surface  forming  a 
larger  effective  drop),  and  why  air  bubbles  promote  ebullition  (the  equilib- 
rium pressure  of  the  vapor  within  them  being  less  than  that  corresponding 
to  the  existing  temperature).  It  is  to  be  noted  that  although  evaporation 
is  promoted  by  a  convex  surface,  boiling  is  not,  the  inclosed  bubbles  being 
subjected  to  a  total  pressure  of  P  +  2  Tl  r,  P  being  the  external  pressure. 
(See  section  158.) 

167.  Daltori s  law  applies  to  mixtures  of  saturated  vapors  and  gases 
when  they  exert  no  chemical  action  on  each  other,  so  long  as  the  pressure 
is  not  very  great.  The  pressure  of  aqueous  vapor,  for  example,  is  not 
appreciably  greater  in  a  vacuum  than  in  air  at  ordinary  pressure.  The 
process  of  vaporization  is  somewhat  delayed  by  air,  diffusion  being 
hindered. 

168.  Fog)  Clouds^  Rain,  Dew. — If  a  superheated  vapor  be  cooled 
below  its  point  of  saturation,  condensation  will  occur.  When  a  mass  of 
air  is  cooled  by  passing  from  above  warm  ocean  currents  to  colder  land 
or  water,  or  from  heated  land  to  cold  water,  or  by  expanding  when  it 
rises  to  regions  of  lower  pressure,  fogs  or  rain  are  caused  by  the  precipi- 
tation of  the  suspended  water  vapor.  The  formation  of  fogs  is  aided  by 
suspended  particles  of  dust  or  smoke,  which  act  as  nuclei  of  condensation. 
This  is  the  principal  cause  of  the  dense  fogs  of  London.  Clouds  are  formed 
in  the  same  way  as  fogs,  but  at  greater  altitudes.  The  particles  of  con- 
densed vapor  slowly  descend,  their  progress  being' impeded  by  viscosity;  but 
as  they  reach  warmer  air  strata  they  again  vaporize,  so  that  a  cloud  appears 
stationary,  although  constantly  falling  and  reforming.  When  a  great  quan- 
tity of  water  vapor  is  in  suspension  and  the  cooling  very  great,  the  particles 
form  drops  and  fall  as  rain.  Mountain  ranges  promote  rainfall  on  the  ocean 
side  in  two  ways:  1,  If  they  are  colder  than  the  air  coming  from  the 
ocean  the  water-vapor  is  condensed;  2,  the  deflection  of  the  air-currents 
upward  to  regions  of  lower  pressure  causes  expansion,  followed  by  cool- 
ing and  precipitation  (see  section  135).  The  climate  on  the  inland  side  is 
usually  dry.  It  was  once  supposed  that  dew  was  a  gentle  rain,  but  Dr. 
Wells,  a  London  physician,  who  carefully  studied  the  matter  about  1818, 
showed  this  to  be  a  mistake.  He  pointed  out  that  dew  was  most  abun- 
dant on  clear  nights;  that  some  classes  of  objects  would  be  covered  by 
dew,  while  neighboring  objects  would  be  dry,  and  that  the  objects  collect- 
ing most  dew  were  invariably  those  having  great  radiative  power  and  low 
specific  heat — that  is,  those  objects  which  cool  most  rapidly  at  night. 
These  facts  show  that  it  is  the  result  of  local  precipitation  on  objects 
cooled  below  the  point  of  saturation  of  the  surrounding  space.  Screens, 
smoke,  and  clouds  hinder  the  formation  of  dew  by  preventing  radiation 
into  free  space,  and  wind  by  preventing  the  air  from  remaining  in  contact 
with  the  objects  long  enough  to  be  cooled  below  the  point  of  saturation — 
"dew-point"  Grass  has  great  radiative  power,  thus  collecting  quantities 
of  dew.  A  thermometer  placed  on  grass  may  indicate  a  temperature  10° 
lower  than  one  a  few  feet  above  it.  The  formation  of  dew  is  illustrated 
by  the  condensation  on  pitchers  containing  cold  water.  (1^ 

169.     Hygrometry  is  the  measurement  of  the  amount  of  moisture  in 

(1)  See  works  on  Meteorology  by  Davis,  Greeley,  Waldo,  Russell. 


ISOTHERMAL    CURVES.  85 

the  atmosphere.  The  absolute  humidity,  or  the  mass  of  water  vapor 
present  in  a  given  space,  is  determined  by  drawing  a  known  volume  of 
air'  through  calcium  -chloride  tubes  and  noting  the  increase  of  weight. 
Our  perception  of  moisture  in  the  atmosphere  does  not  depend,  however, 
upon  absolute  humidity,  but  upon  relative  humidity*  which  is  measured  by 
the  ratio  of  the  pressure  of  the  vapor  present  to  that  which  accompanies 
saturation  at  the  same  temperature.  The  dew-point  may  be  determined 
with  various  forms  of  hygrometer,  the  best  of  which  is  Regnault's.  A 
polished  silver  tube  contains  ether  and  a  thermometer.  The  ether  is 
caused  to  evaporate  by  blowing  air  through  it,  and  the  temperature  noted 
at  which  dew  forms  on  the  tube  and  when  it  disappears.  The  mean  is  the 
dew-point.  If  p  is  the  saturation  pressure  corresponding  to  this  tempera- 
ture and  P  that  corresponding  to  the  temperature  of  the  atmosphere,  the 
relative  humidity  is  pjP.  The  wet  and  dry  bulb  thermometer  is  also 
much  used.  One  thermometer  has  a  wet  cloth  around  its  bulb  and  is 
cooled  by  evaporation,  the  rate  of  which  depends  on  the  amount  of 
aqueous  vapor  present.  The  amount  of  aqueous  vapor  is  then  determined 
by  an  empirical  calibration. 

Sultry  and  oppressive  weather  is  due  to  great  humidity,  preventing 
evaporation  of  perspiration. 

170.  Vapor  Density  —  The  density  of  a  vapor  is,  strictly  speaking, 
the  mass  of  unit  volume  under  standard  conditions.  The  name  is  often 
applied,  however,  to  the  specific  gravity  of  a  vapor,  with  dry  air  at  same 
pressure  and  temperature  taken  as  the  standard.  If  the  density  of  the  air 
at  0°  and  76  cm.  pressure  is  PO  under  other  conditions  — 

(125)  P 


7760 

7"  being  the  absolute  temperature.  In  the  various  methods  of  determin- 
ing vapor  density  a  vessel  of  known  volumn  is  filled  with  superheated 
vapor  and  its  mass  m  determined  ;  then  D  =  vpjm.  This  gives  the  relative 
vapor  density  at  the  temperature  and  pressure  of  the  experiment.  If  the 
vapor  obeys  the  laws  of  Boyle  and  Charles  (which  most  vapors  approxi- 
mately do)  above  the  saturation-point,  the  same  value  holds  for  all  other 
conditions.  The  density  of  water  vapor  is  about  $/%  that  of  air,  so  that  it 
is  necessary  to  make  corrections  for  the  hygrometric  state  in  hysometrical 
observations. 

PROBLEMS. 

77  .yu-What  is  the  weight  of  a  liter  of  water  vapor  at  20°  and  at  10  cm.  pressure  ? 

78.  The  dew  point  of  air  at  18°  and  76  cm.  pressure  is  5°.     What  is  the  weight 
of  water  vapor  per  liter  ? 

79.  At  what  temperature  will  water  boil  on  Mount  Tamalpais  (3000  feet)  ;  on 
Mount  Whitney,  (15,000  feet)? 

171.  Isothermal  Curves.  —  The  locus  of  the  points  of  intersection 
of  rectangular  coordinates  representing  pressures  and  volumes  (tempera- 
tures remaining  constant)  is  called  an  isothermal  curve.  For  a  perfect 
gas,  Boyle's  law  shows  us  that  the  isothermals  would  be  rectangular 
hyperbolas  ;  for  saturated  vapors  they  are  straight  horizontal  lines  (  /  con- 
stant) ;  for  unsaturated  vapors  they  approximate  rectangular  hyperbolas  ; 
for  solids  and  liquids  they  are  nearly  vertical  lines  (v  nearly  constant). 


86  HEAT. 

V 

A  set  of  isothermals  showing  the  pv  relations  for  a  wide  range  of  tempera- 
tures gives  us  a  complete  history  of  the  conduct  of  the  substance  with 
regard  to  changes  of  state. 

(Lines  of  equal  temperature  on  the  earth's  surface  are  also  called 
isothermals). 

172.  Continuity  of  the  Liquid  and  the  Gaseous  States — 
Critical  Temperature. — In  1822  Cagniard  de  la  Tour  heated  ether, 
alcohol,  and  other  liquids  in  sealed  glass  tubes  to  very  high  temperatures. 
Under  such  circumstances  ebullition  is  impossible,  as  any  bubbles  formed 
within  the  mass  are  exposed  to  a  pressure  greater  than  that  of  the  vapor 
within  them.  There  was  a  gradual  evaporation,  and  at  a  certain  tempera- 
ture (about  200°  for  ether,  259°  for  alcohol)  the  meniscus  suddenly 
disappeared,  leaving  the  tube  apparently  filled  with  a  homogeneous  mass. 
De  la  Tour  concluded  that  the  liquid  was  entirely  vaporized. 

In  18.34  Thilorier  liquified  carbon  dioxide  by  cold  and  pressure,  and 
in  1863  Dr.  Andrews  plotted  its  isothermal  curves  through  a  wide  range 
of  temperature.  The  gas  was  compressed  in  a  narrow  glass  tube  by  a 
column  of  mercury,  which  also  communicated  with  an  air  manometer 
registering  the  pressure.  These  curves  throw  great  light  on  the  nature  ot 
gases  and  liquids.  At  13°.  1  (ordinary  temperature)  the  substance  dimin- 
ishes in  volume  almost  inversely  as  the  pressure.  When  the  latter  reaches 
about  50  atmospheres  it  remains  constant  until  the  volume  is  reduced  to 
about  one-fifth,  after  which  increase  of  pressure  produces  little  cha  nge  in 
volume.  At  50  atmospheres  the  gas  begins  to  condense  to  a  liquid,  this 
being  the  saturation  pressure  corresponding  to  that  temperature.  The 
curve  for  21°.  5  likewise  has  two  sharp  discontinuties  and  a  horizontal 
portion,  but  that  for  31°.  1  is  quite  different.  It  shows  two  inflections,  but 
there  is  no  point  at  which  the  curve  is  horizontal,  as  would  be  the  case 
with  a  saturated  vapor.  At  a  pressure  of  about  73-75  atmospheres  the 
meniscus  disappeared  as  in  De  la  Tour's  experiments.  The  curves  above 
31°. 5  show  less  evidence  of  irregularity,  and  that  for  48°.  1  does  not 
differ  greatly  from  the  isothermal  of  an  almost  perfect  gas  such  as  air. 
Andrews  concluded  that  above  the  temperature  of  33°,  which  he  called 
the  critical  temperature,  carbon  dioxide  cannot  exist  as  a  liquid.  Later 
researches  by  Amagat  give  the  critical  constants  for  this  gas  as  31°.  35 
and  72.9  at.  Andrews  investigated  other  gases  in  the  same  manner 
with  similar  results.  He  inferred  that  there  was  "continuity  of 
state,"  or  gradual  passage  from  a  liquid  to  a  gas.  About  1870 
Cailletet  and  Collardeau  studied  the  subject  and  concluded  that  the 
disappearance  of  the  meniscus  did  not  prove  that  no  liquid  persisted. 
Iodine  dissolves  in  liquid  carbon  dioxide,  but  not  in  the  gas ,  and  they 
showed  that  on  partly  liquefying  the  gas  in  a  tube  containing  iodine  the 
latter  was  dissolved  in  the  liquid ;  on  heating  above  the  critical  tempera- 
ture, the  line  of  demarcation  remained  unchanged.  They  concluded 
that  liquid  was  still  present,  although  its  surface  tension  had  vanished. 
It  seems  evident,  however,  that  above  the  critical  temperature  the  sub- 
stance has  properties  different  from  those  of  either  a  liquid  or  a  gas.  It 
is  possible  to  change  a  substance  from  an  undoubtedly  gaseous  to  an 
undoubtedly  liquid  state  without  any  visible  change.  If  carbon  dioxide 
is  heated  to  50°,  compressed  under  100  atmospheres  pressures,  then 
cooled  under  constant  pressure  to  25°,  it  is  certainly  a  liquid  ;  but  no 
visible  liquefaction  has  occurred. 

Surface  tension  diminishes  as  the  temperature  increases,  vanishing  at 


LIQUEFACTION    OF   GASES.  87 

the  critical  temperature.  This  means  that  the  forces  between  the 
molecules  of  the  liquid  and  those  of  the  gas  are  the  same.  The  latent 
heat  of  vaporization  also  vanishes,  for  the  density  of  the  saturated  vapor 
approaches  that  of  the  liquid  near  the  critical  point;  so  that  no  work  is  to 
be  done  in  separating  the  molecules.  Regnault's  equation  (123)  indicates 
that  the  latent  heat  of  water  vanishes  at  706°,  while  its  critical  tempera- 
ture is  about  365°;  but  the  formula  was  deduced  from  observations 
within  a  limited  range  and  cannot  be  expected  to  hold  for  high  tem- 
peratures. 

Distinction  Between  Gas  and  Vapor. — A  gas  may  be  defined  as  a 
vapor  above  its  critical  temperature;  a  vapor  as  a  gas  between  its  critical 
temperature  and  the  point  of  liquefaction.  A  vapor  shows  very  marked 
deviations  from  Boyle's  law,  but  van  der  Waals'  equation  holds  fairly 
well,  giving  curves  resembling  those  of  carbon  dioxide  above  the  critical 
temperature.  Superheated  water  vapor  approximately  obeys  the  law 
^17/16  =  constant  (Rankine). 

173.  Liquefaction  of  Gases.— Faraday,  between  1823  and  1849, 
liquefied  all  the  gas  known  to  him  except  hydrogen,  nitrogen,  oxygen, 
carbon  monoxide,  nitrous  oxide,  and  methane.  His  method  was  to  place 
the  gas  in  solution,  or  a  substance  from  which  the  gas  could  be  evolved  by 
heat,  in  one  arm  of  a  sealed  U-tube,  place  the  other  arm  (inverted)  in  a 
freezing  mixture,  and  evolve  the  gas  by  applying  heat,  thus  producing 
great  pressure.  The  gas  condensed  in  the  cooled  arm.  It  is  evident 
from  Andrews'  work  that  in  order  to  liquefy  a  gas  it  is  necessary  to 
reduce  it  below  the  critical  temperature.  The  method  adopted  is  based 
upon  the  principle  of  freezing-machines.  About  1877  Cailletet  in  Paris 
and  Pictet  in  Geneva  both  succeeded  in  liquefying  oxygen,  nitrogen,  and 
carbon  monoxide.  In  Pictet' s  apparatus  liquid  sulphur  dioxide  was  allowed 
to  evaporate  under  low  pressure  around  a  tube  containing  carbon  dioxide 
under  pressure.  The  latter  was  thus  liquefied  at  a  temperature  of — 70°. 
This  was  allowed  to  evaporate  around  another  tube  containing  oxygen 
under  275  atmospheres  pressure,  and  the  latter  liquefied,  at  a  tempera- 
ture of  — 130°.  Still  lower  temperatures  were  produced  by  Wroblewski 
and  Olszewski,  who  began  their  investigations  together  about  1880,  but 
finally  worked  independently.  They  used  apparatus  similar  in  principle 
to  that  of  Pictet,  but  secured  lower  temperatures  by  evaporating  liquid 
ethylene  ( — 152°),  the  resulting  liquid  oxygen  evaporating  at  low 
pressure  condensed  nitrogen  at  — 200°,  the  latter  evaporating  under  low 
pressure  produced  a  temperature  estimated  at  — 225°  and  froze.  Up 
to  1895  hydrogen  alone  remained  unliquefied,  although  Pictet  and 
Wroblewski,  by  using  the  greatest  cold  and  pressure  obtainable  and 
allowing  it  to  expand  suddenly,  had  produced  a  temporary  mist.  In 
1895  Olszewski  reduced  this  gas  to  the  lowest  temperature  obtainable  by 
the  evaporation  of  liquid  nitrogen,  and,  allowing  it  to  expand  suddenly, 
observed  evidences  of  ebullition.  Dewar,  of  the  Royal  Institution,  has 
recently  done  much  work  in  liquefying  gases.  In  1898  he  succeeded  in 
liquefying  hydrogen.  He  found  the  constants  given  for  this  gas  in  the 
table  below.  When  liquid,  its  density  was  one-fourteenth  and  its  surface 
tension  one-thirty-fifth  that  of  water.  At  — 258°  it  froze  to  a  solid,  with 
a  coefficient  of  expansion  ten  times  that  of  gas,  and  a  specific  heat  five 
times  that  of  water.  The  solid  hydrogen  in  a  vacuum  is  reduced  by 
evaporation  to  a  temperature  of  about  — 260°  or  13°  absolute.  It  is 
an  ice-like  solid  having  a  density  one-eleventh  that  of  water. 


88  HEAT. 

.  .  Boiling  P.  at  Freezing-  Density 

-nt-   1  •  Cnt-  *  •  atmos.  P.  point.  at  B.  P. 

H —246°  15  —252°  —258°  0.070 

N —146°  35  —194°  —214°  0.885 

O —115°  50.8  —  181°.4  1.124 

Air —140°  39  —  191°.4  

Argon —121°  50.6  —187°  —189°. 6  

CO2 -f    31°  73  —    80° 

Ether -f  194°  35.6  +    35°  

All  the  above  when  liquefied  are  colorless  except  oxygen,  which  is 
light  blue. 

These  temperatures  are  to  be  regarded  only  as  approximations.  They 
were  determined  either  from  a  hydrogen  or  helium  thermometer  (which 
probably  deviates  widely  from  Charles'  law  at  these  low  temperatures)  or 
by  a  thermopile  calibrated  by  a  hydrogen  thermometer. 

Liquid  air  is  now  generally  made  by  a  purely  mechanical  regenerative 
process,based  on  the  fact  referred  to  in  Sec.  13y  thalfj^ases  on  expanding 
againsrvpressure  become  cooled.  Air  is  first  compressed  by  a  steam 
engine,  and  is  then  allowed  to  expand  into  a  second  reservoir,  after  being 
cooled  by  passing  through  a  cold-water  jacket.  In  expanding  it  is  further 
cooled ;  it  is  then  pumped  back  into  the  first  reservoir,  through  a  pipe 
concentric  with  and  surrounding  that  through  which  it  entered  the  second 
reservoir.  It  then  cools  the  second  installment  of  air  to  a  lower  tempera- 
ture that  it  had  itself ;  and  thus  by  successive  steps  the  air  is  cooled  until 
a  part  of  it  becomes  liquefied.  This  method  was  first  applied  by  Linde, 
of  Munich. 

Helium  is  the  only  known  gas  which  has  not  been  liquefied.  Dewar 
believes  that  its  critical  temperature  lies  near  5°  absolute. 

References. —  Dewar,  Scientific  Uses  of  Liquid  Air,  London  Electrician,  July 
12,  1895  ;  Presidential  Address  at  British  Association,  Nature,  Vol.  66,  p.  462,  1902, 
and  Chemical  News,  Vol.  86,  p.  127,  1902  ;  Harden,  Liquefaction  of  Gases  ; 
Sloane,  Liquid  Air.  For  a  description  of  Linde's  machine,  see  Watson's  Physics. 

OTHER  EFFECTS  OF  HEAT. 

174.    Changes  in   Molecular  Constraints  and   Structure. — 

Certain  continuous  changes,  less  sharply  defined  than  those  of  state,  are 
produced  by  heat.  As  temperature  rises — 

Elasticity  diminishes.  The  percentage  diminution  in  Young's  modulus 
between  0°and  100°  is:  Platinum,  .89;  iron,  2.35;  brass,  4.4;  aluminum, 
19.5.  Dewar  observed  a  great  increase  of  elasticity  and  strength  in  metals 
at  the  low  temperatures  produced  by  evaporating  liquid  air. 

The  viscosity  of  liquids  diminishes,  owing  to  greater  molecular  free- 
dom ;  for  water  at  50°  it  is  less  than  one-third  that  at  0°. 

The  viscosity  of  gases  increases  from  the  same  cause,  since  the  motion 
of  one  stratum  over  another  is  retarded  by  interchange  of  molecules. 

Surface  tension  diminishes.  According  to  the  results  of  Frankenheim, 
if  h  be  the  height  in  a  capillary  tube  at  0° — 

(126)  h>-=h0(l-ct). 

Some  values  for  c  are:  Water,  0.0019;  alcohol,  0.0024;  ether, 
0.0047.  Assuming  these  results  to  hold  for  any  temperature  (which  is 
not  quite  legitimate,  as  they  were  deduced  between  0°  and  100°),  the 
critical  temperature  =  1  /  c  would  be  :  Water,  526°;  alcohol,  417°;  ether, 
213°,  which  shows  a  close  agreement  with  facts  only  in  the  case  of  ether. 


CHEMICAL    COMBINATION    AND    DISSOCIATION.  89 

Gradual  or  abrupt  changes  in  molecular  structure  may  be  caused  by 
heat.  Sulphur  heated  beyond  250°  and  suddenly  cooled  by  pouring  in 
water  becomes  a  viscous  tenacious  mass.  It  slowly  returns  to  its  normal 
form,  or  if  heated  to  100°  it  returns  suddenly  with  the  evolution  of  heat. 
If  iron  is  heated  to  white  heat  and  allowed  to  cool  slowly,  at  a  dull  red  it 
begins  to  glow  more  brightly,  showing  an  evolution  of  heat  due  to  some 
internal  change  (recalescence). 

175.  Chemical  Combination  and  Dissociation. —  Often  chem- 
ical   combination  takes  place  at  a  high  temperature.     At  a   still  higher 
temperature  dissociation  may  result,  as  in  the  case  of  iodine,  sulphur  and 
other  elements.     This  is  shown  by  a  gradual  decrease  in  the  density  due 
to   the   splitting   of  molecules   into   atoms.     According   to   the  modern 
theories  a  fraction  of  the  molecules  of  a  gas  or  an  electrolyte  in  solution  is 
always  dissociated,  the  proportion  increasing  with  the  temperature.     At 
any     given    temperature,    recombination    balances    dissociation    due    to 
impact,  etc.,  producing  a  state  of  kinetic  equilibrium  as  in  evaporation.     In 
this  way  abnormal  gaseous  and  osmotic  pressures,  lowering  of  the  freezing- 
point,   raising  of  the  boiling-point,  and  electrolytic   conduction   can   be 
explained. 

Latent  Heat  of  Combination. — Dissociation  involves  work  done  upon 
molecules,  or  an  absorption  of  heat ;  combination,  work  done  by  them  in 
falling  together,  or  an  evolution  of  heat.  The  heat  in  calories  evolved  by 
one  gram  of  some  substances  burning  in  oxygen  is  as  follows  :  Hydrogen, 
34,000;  marsh  gas,  13,000;  ether,  9,000;  coal,  8,000;  wood,  4,000; 
iron,  1,500.  These  figures  area  measure  of  thermal  efficiency,  but  not  of 
economic  efficiency,  which  depends  on  prices.  In  the  transformation  of 
oxygen  and  hydrogen  to  one  gram  of  ice,  the  amount  of  heat  given  out  is 
3,777  +  536  +  80  =  4,393  calories,  and  the  absorption  of  the  same 
amount  will  return  the  elements  to  their  original  state. 

All  the  effects  of  heat  thus  far  described  are  in  accordance  with  and  are 
best  explained  by  the  dynamical  theory  of  heat. 

176.  Electrical  Effects. —  If  a  circuit  be  made  of  two  or  more 
different  metals  soldered  together,  any  difference  of  temperature  between 
the  junctions    will   cause  a  flow  of  electricity,    called  a  thermo-electric 
current.      If  the   temperature    of  an   electrical   conductor   changes,    its 
resistance  also  changes,  usually  increasing  with  the  temperature. 

177.  Measurement  of  High  and   Low  Temperatures.— The 

expansion  of  a  substance  (Wedgewood  pyrometer)  is  a  doubtful  measure 
of  high  temperature  on  account  of  the  unknown  rate  of  expansion ;  so  are 
methods  depending  on  specific  heat.  Violle  determined  the  temperature 
of  the  electric  arc  by  knocking  a  button  of  carbon  from  the  hot  portion 
into  a  calorimeter  and  calculating  the  original  temperature  from  the 
specific  heat  and  change  in  temperature.  The  specific  heat  of  carbons  at 
high  temperatures  is  unknown;  consequently  the  result,  3,500°,  is 
uncertain.  This  is  probably  the  highest  temperature  attained  by  us, 
except,  perhaps,  that  of  the  electric  spark.  The  most  accurate  methods 
for  very  low  or  very  high  temperatures  are  gas  thermometers  in  porcelain 
bulbs ;  platinum  thermometers  based  on  changes  of  electric  resistance, 
which  read  accurately  up  to  1,300°;  thermo-electric  couples  of  platinum  - 
iridium,  with  which  Barus  read  up  to  1,700°. 

Differential  Measurements. — With  a  sensitive  galvanometer  and  a 
thermopile  (thermal  couple)  differences  of  .001°  may  be  detected,  and  by 


90  HEAT. 

Langley's  bolometer,  which  utilizes  the  change  of  resistance  in  a  thin  plati- 
num wire,  differences  as  small  as  .000001°.  Equally  sensitive  is  Boys' 
radiomicrometer,  made  by  suspending  a  thermo-electric  circuit  in  a  mag- 
netic field.  The  most  sensitive  instrument,  however,  is  a  modification  of 
the  Crooke's  radiometer. 


TRANSFER   OF   HEAT. 

Experience  shows  that  if  two  or  more  bodies  are  at  different  temper- 
atures, there  is  a  tendency  toward  equalization  of  temperatures,  whether 
they  are  in  contact  or  not.  The  transfer  of  heat  takes  place  by  three 
distinct  but  generally  cooperating  methods — radiation,  convection,  and 
conduction. 

178.  Radiation. — It  is  found  that  isolated  bodies  at  a  higher  tem- 
perature than  their  surroundings  lose  heat  even   in   a   vacuum.     It   is 
assumed  that  the  vibratory  motion  of  the  molecules  is  imparted  to  the 
surrounding  ether,    and   these  waves,    striking    other  objects,   set  their 
molecules  also  in  vibration.     Although  radiant  energy  is  capable  of  trans- 
formation to  heat,  it  is  not  itself  heat.     The  ether  or  other  transparent 
medium   is    not   warmed  by  it.     Radiation  is  therefore  more   properly 
discussed  under  the  head  of  Light. 

179.  Convection. — If  a  region  within  a  fluid  acquires  a  higher  tem- 
perature   than  that  of  the   surrounding  fluid,    its  density  becomes  less 
(except  in  a  few  cases  such  as  that  of  water  below  4°),  and  the  heated  mass 
rises.     Its  place  is  taken  by  the  colder  fluid  around  it.      By  this  process 
of  convection  circulation  is  thus  established,  motion  being  from  the  colder 
to  warmer  regions  below  and  from  warmer  to  colder  above.     Topler's 
"schlieren"  method  of  visibly  projecting  such  phenomena  on  a  screen  by 
passing  light  from  a  point  source  through  the  region  of  circulation  shows 
these  convection  currents  rising  from  flames  or  warm  objects  in  air  or 
through  fluids   heated  at  the  bottom  of  the  containing  vessel,    and  the 
diffusion  currents  in  solutions. 

Convection  phenomena  play  an  important  part  in  modifying  climate. 
The  heated  air  at  the  equator  rises,  flows  toward  the  poles,  and  descends 
in  temperate  latitudes;  its  place  is  taken  by  colder  air  passing  to  the  equa- 
tor. Such  currents,  modified  in  direction  by  the  rotation  of  the  earth, 
give  rise  to  the  trade  winds.  Cyclones,  tornadoes,  and  lesser  wind  disturb- 
ances are  caused  in  a  similar  way.  The  water  of  the  oceans  behaves  in 
the  same  manner,  expanding  several  feet  higher  than  its  normal  level  at 
the  equator,  and  flowing  down  hill  to  the  poles  in  various  currents,  which 
are  modified  in  their  directions  by  the  conformation  of  continents  and 
islands,  as  well  as  by  the  rotation  of. the  earth.  The  air  above  these  warm 
currents  becomes  healed,  and  may  be  carried  by  the  wind  to  land,  thus 
modifying  temperatures  near  the  coast. 

Convection  has  its  practical  applications  in  ventilation  and  the  heating 
of  houses  by  hot  water  and  air. 

In  the  case  of  convection,  masses  of  heated  fluid  are  transported 
bodily,  carrying  the  heat  with  them.  The  transfer  of  heat  (kinetic 
energy)  from  one  molecule  to  another  is  by  radiation  or  conduction. 

180.  Conduction. — If  a  solid  such  as  a  metal  rod  have  one  end 
inserted  in  a  flame,  that  end  will  rapidly  rise  in  temperature.     The  whole 
rod  will  in  time  become  heated,   but  more  and  more  slowly  and  with  a 


CONDUCTION.  91 

lower  final  temperature,  as  the  distance  from  the  heated  end  increases. 
We  may  imagine  that  the  nearer  molecules  have  their  kinetic  energy 
increased,  and  transfer  a  portion  of  this  energy,  by  impact  or  otherwise, 
to  adjacent  molecules  until  the  effect  is  felt  at  the  other  end  of  the  rod. 
Each  molecule  passes  on  only  a  fraction  of  its  gain  in  energy,  so  that  the 
temperature  diminishes  in  going  from  the  source.  After  a  time  the  loss  at 
a  given  point  due  to  radiation  and  conduction  balances  the  gain,  and  a 
condition  of  steady  flow  is  reached.  If  a  curve  be  plotted  with  tempera- 
tures and  distances  along  the  rod  as  coordinates,  and  \idt  and  dx  be  small 
increments  of  /  and  x,  the  ratio  dt  /  dx  =  tan«  is  called  the  temperature 
gradient. 

Therrnometric  Conductivity. — If  uniform  rods  of  different  materials  be 
covered  with  wax  and  exposed  to  the  same  source  of  heat,  the  wax  will 
melt  at  different  rates — rapidly  on  the  metal  rods  and  very  slowly  or  not 
at  all  on  wood  and  glass  rods.  The  attainment  of  a  given  temperature  at 
a  given  point  depends  both  on  the  amount  of  heat  reaching  that  point  and 
the  specific  heat  of  the  substance. 

Thermal  Conductivity  is  defined  as  the  number  of  heat  units  which 
flow  per  second  through  a  unit  cube  of  the  substance  when  its  opposite 
faces  are  maintained  at  a  temperature  difference  of  1°.  Experiment  shows 
that  the  quantity  is,  within  small  limits,  directly  proportional  to  the  tem- 
perature difference  and  inversely  to  the  thickness.  Considering  a  slab  of 
area  A  and  thickness  dx,  with  a  small  temperature  difference  dt,  if  K  be 
the  thermal  conductivity  and  H  the  amount  of  heat  flow  per  second, 

(127)  H=KA^> 

A  similar  equation  applies  to  diffusion,  osmotic  pressure  taking  the 
place  of  temperature.  Ohm's  law  for  electric  flow  is  also  of  the  same 
form. 

Some  values  of  Km  C.  G.  S.  units  are:  Copper,  0.96;  iron,  0.16; 
lead,  0.11;  sandstone,  0.003;  water,  0.0015;  wood,  0.0004  (the  values 
parallel  to  the  grain  and  at  right  angles  to  it  are  different) ;  flannel,  0.00023; 
air,  0.00005;  hydrogen,  0.00033. 

If  k  is  the  thermometric  conductivity,  p  the  density,  and  s  the  specific 
heat,  A"=  psk.  For  iron  ^  =  0.86,  for  lead,— 0.35;  consequently  k\\  k\= 
16/87  : 11/35,  or  the  rate  at  which  lead  is  heated  by  conduction  is  greater 
than  that  of  iron,  although  its  thermal  conductivity  is  less. 

The  conductivity  of  crystals  differs  along  different  axes  is  shown  by 
touching  a  hot  needle  to  a  layer  of  wax  on  a  quartz  crystal  cut  parallel  to 
the  principal  axis.  The  wax  will  melt  in  elliptical  shape,  with  axes  in  the 
ratio  289  :  158. 

The  conductivity  of  liquids  (except  melted  metals)  and  gases  is  usually 
small  and  difficult  to  determine  on  account  of  disturbances  due  to  convec- 
tion and  radiation.  Hydrogen  has  a  conductivity  about  seven  times  as 
great  as  air,  as  shown  by  its  cooling  effect  on  a  wire  heated  by  electricity. 

It  is  probably  impossible  to  determine  the  pure  conductivity  (transfer 
of  energy  by  collision)  in  gases,  because  of  simultaneous  diffusion  and 
convection.  The  value  of  flannel  as  clothing  is  shown  by  its  poor  conduc- 
tivity, which  is  largely  due  to  the  imprisoned  air.  Double  walls,  with  the 
space  between  entirely  exhausted,  form  a  still  better  screen,  as  heat  can 
then  escape  only  by  radiation.  In  order  to  protect  liquid  air  from  heat, 


92  HEAT. 

Dewar  devised  glass  test-tubes  and  flasks  with  double  walls,  with  the  inter- 
space exhausted  to  prevent  conduction,  and  silvered  outside  to  reflect 
incident  radiation.  This  reduced  the  loss  to  about  one-fifth  what  it  would 
otherwise  have  been. 

A  flame  will  not  penetrate  the  meshes  of  wire  gauze,  because  its  heat 
is  so  rapidly  conducted  away  that  the  temperature  of  combustion  cannot 
be  maintained.  This  fact  is  utilized  in  Davy's  safety  lamp,  which  can  be 
used  in  the  inflammable  gases  found  in  mines. 

Owing  to  the  small  conductivity  of  heat  by  the  soil,  the  maxima  and 
minima  of  temperature  below  ground  occur  much  later  than  at  the  surface, 
and  the  fluctuations  rapidly  get  less  and  less.  The  daily  wrave  penetrates 
2  or  3  feet,  the  yearly  wave  40  or  50  feet,  in  temperate  latitudes.  At 
greater  depths  the  temperature  is  approximately  constant. 

QUESTIONS. 

80.  Can  any  amount  of  pressure  liquefy  argon  at  a  temperature  of  —100°  ? 

81.  What  conditions  will  prevent  smoky  chimneys? 

82.  In  heating  houses  by  hot  air  or  water,  would  it  be  satisfactory  to  place  the 
furnace  on  the  top  floor? 

83.  How  does  a  chimney  assist  ventilation  ? 

84.  Why  does  a  lamp  chimney  make  the  flame  brighter  and  free  from  smoke  ? 

85.  If  a  room  has  a  single  window  (ordinary  double-sash)  how  must  it  be 
opened  to  secure  best  ventilation  ? 

86.  Why  do  winds  usually  blow  down  mountain  canons  at  night  ? 

87.  The  prevailing  winds  blow  from  the  ocean  in  California  and  Western 
Europe,  from  the  land  in  the  North  Atlantic  States.     How  is  the  climate  of  these 
regions  affected  thereby  ? 

88.  Why  does  so  little  rain  fall  between  the  Sierras  and  the  Rocky  Mountains  ? 

89.  Which  feels  colder  to  the  touch,  copper  or  iron,  when  both  are  at  the 
temperature  of  the  body  ? 

90.  State  four  objections  to  the  use  of  water  as  a  thermometric  substance. 


THERMODYNAMICS. 

181.  The  two  fundamental  principles  of  Thermodynamics,  or  the 
study  of  the  mechanical  effects  of  heat,  are  based  on  the  following  experi- 
mental facts  :  (1)  When  no  other  forms  of  energy  are  involved,  the 
performance  of  a  definite  amount  of  mechanical  work  by  a  heat  engine  is 
accompanied  by  the  disappearance  of  a  definite  amount  of  heat — and  vice 
versa;  (2)  A  heat  engine  can  be  continuously  operated  only  by  the 
passage  of  heat  from  a  hotter  to  a  colder  body — heat  cannot  run  "up 
hill."  About  1824  Sadi  Carnot  laid  the  foundation  of  the  theory.  By 
supposing  any  material  substance  to  undergo  a  cycle  of  changes  in 
pressure  and  volume  by  the  application  of  heat,  finally  returning  to  its 
original  condition,  thus  eliminating  any  energy  losses  due  to  change  of 
condition,  he  sought  to  obtain  a  definite  relation  between  heat  and  \vork, 
although  at  the  time  he  held  the  material  idea  of  heat.  We  may  use  his 
process,  remembering;  the  equivalence  of  heat  and  energy.  In  order  to 
do  this  we  must  consider  two  kinds  of  change — isothermal,  in  which  the 
substance  remains  at  constant  temperature,  and  adiabatic,  in  which  no 
heat  is  given  to  or  taken  from  it,  the  substance  being  cooled  or  heated  in 
doing  work  or  having  work  done  on  it. 


CARNOT'S  REVERSIBLE  CYCLE  —  ABSOLUTE  TEMPERATURE.         93 

182.  Carnot's  Reversible  Cycle.  —  The  substance  may  be  sup- 
posed to  be  contained  in  a  non-conducting  cylinder,  with  perfectly 
conducting  bottom,  and  provided  with  a  piston.  Place  the  cylinder  on  a 
non-conducting  support  and  compress  the  substance  adiabatically  from 
A  to  B,  its  temperature  rising  from  T0  to  Z1,.  Allow  it  to  expand 
isothermally  to  C,  keeping  its  temperature  constant  by  placing  it  on  a 
source  at  temperature  Tt.  The  heat  absorbed  from  this  source  is  H^. 
Place  it  again  on  the  non-conducting  stand  and  compress  it  adiabatically 
until  it  comes  to  its  original  temperature  T0  at  D.  Compress  it 
isothermally  to  A,  the  heat  given  to  a  refrigerator  at  temperature  TQ 
being  H0.  The  external  work  done  by  the  substance  is  W=  Hr  —  H0 
(//'being  measured  in  ergs),  for  since  it  is  in  its  original  condition  the 
total  internal  work  must  be  zero.  The  efficiency  is 

(128)          Wj  //;  =  (//,  -  /O  /  H,  =  (7;  -  T0)  I  rx. 


The  external  work  done  is  evidently  equal  to  the  area  A  BCD 
(indicator  diagram)  \{  p  and  v  are  given  in  C.  G.  S.  units.  If  the  cylinder 
is  always  at  the  same  temperature  as  the  source  or  the  refrigerator  when 
in  contact  with  them,  it  is  evident  that  the  process  is  reversible;  the  work 
W  being  done  on  the  substance,  H0  units  of  heat  are  taken  from  the 
refrigerator  and  H^  units  returned  to  the  source.  This  imaginary  arrange- 
ment is  called  Carnot's  perfect  reversible  engine.  No  heat  is  supposed  to 
be  lost  by  friction,  etc.  All  such  engines  must  be  equally  efficient,  what- 
ever the  working  substance,  else  by  running  one  with  another  of  different 
efficiency  a  surplus  of  useful  work  might  be  indefinitely  obtained,  which 
may  be  shown  to  involve  a  continuous  transfer  of  heat  from  the  refrig- 
erator to  the  source  at  higher  temperature,  in  opposition  ~  to  one  of  the 
fundamental  principles  of  thermodynamics. 

183.  Absolute  Temperature.  —  Thomson  (Lord  Kelvin)  adopted 
an  absolute  scale  of  temperature  based  on  the  assumption  that  in  a  perfect 
engine  the  temperatures  of  the  source  and  the  refrigerator  are  proportional 
to  the  amounts  of  heat  absorbed  from  the  one  and  returned  to  the  other, 
or  TJHj_  =  TJH0.     If  the  thermodynamic  substance  be  a  gas  obeying 
Boyle's  and  Charles'  laws,  it  may  be  shown  that  the  absolute  temperature 
as  determined  from  the  expansion  of  the  gas  coincides  with  that  defined 
by  the  above  relation. 

Equation  (128)  shows  that  efficiency  depends  only  on  the  temper- 
atures of  the  source  and  the  refrigerator,  and  no  engine  could  be  perfectly 
efficient  unless  the  refrigerator  be  at  absolute  zero.  Evidently  no  lower 
temperature  is  possible.  Ether  has  a  smaller  latent  heat  than  water,  boils 
at  a  lower  temperature,  and  has  a  higher  vapor  pressure  ;  yet  it  has  no 
advantage  over  water,  because  it  cools  so  much  more  rapidly  on  expan- 
sion (thus  reducing  the  pressure)  that  all  its  other  advantages  are 
neutralized. 

184.  Change    of    Freezing-point    Determined    by    Cyclic 
Process.  —  This  is  an  instructive  application  of  Carnot's  method.       Let 
AB  represent  the  change  of  volume  of  m  grams  due  to  change  of  state, 
the  temperature  and  pressure  both  remaining  constant.     If  z\  and  z>2  are 
the  volumes  of  unit  mass  in  the  first  and   second   states  respectively, 
m  =  AB'iiy^  —  z/r)  and  the  heat  obsorbed  is  — 

(129)  H=  Lm  =  LAB!(v2  -  z/f). 


94  HEAT. 

If  the  temperature  be  lowered  adiabatically  to  T,  and  the  mass  m 
condensed  by  pressure,  the  heat  evolved  may  be  represented  by  a  similar 
equation  ;  or,  more  simply,  calculated  from  the  efficiency  equation  which 
gives  W=  H(T2~-  T^)/T2;  also,  since  AB  CD  is  approximately  a  par- 
allelogram, W=ABI(p2—  />r),  which  gives  — 


(130)  7;-rI  =  (n- 

This  expression  is  negative  for  v2  <vlt  and  positive  for  v2>vt  ;  hence  the 
freezing-point  is  lowered  in  the  first  case,  raised  in  the  second,  by  increase 
of  pressure.  The  lowering  of  the  freezing-point  of  water  calculated  from 
this  expression  is  .0075;  the  observed  value  is  .0072  (see  section  149). 
This  equation,  of  course,  applies  also  to  vaporization,  in  which  case 
Z'2  —  vl  is  always  positive. 

185.    Adiafoatic  Expansion  of  Gases.  —  When  heat  is  applied  to 

a  gas,  if  dT  and  dv  are  the  small  changes  in  T  and 


(131)  H= 

This  is  the  algebraic  expression  of  the  first  law  of  Thermodynamics.     In 
case  of  adiabatic  expansion  this  expression  equals  zero;  also  pv=  mRT 

and  (p  +  dp)  (v  +  dv)  =  mR  (T+dT),  from  which  dT=P  '  dv  +  '*  '  dp  . 

m  JT\. 

From  this  and  the  relation  <rp  —  cv  =  R  //we  get  by  substitution  in  (131)> 

(132)  ^  +  k  ^  =  0  (putting  k  =  c,  I  O- 

Assume  the  adiabatic  relation  of  pressure  and  volume  to  be  pvn  ~ 
constant  =  (p  +  dp)  (v-{-  dv)n  =pvn  +  npv*~ldv  +  v  .  dp  (neglecting  the 
vanishing  product  of  infinitesimals),  from  which  — 

(133)  ^.  +  w*L  =  o. 

p  v 

Comparing  with  (132)  we  find  the  relation  to  be  — 

(134)  pv*=C 

Elasticity  is  stress  I  strain  or  v  .  dp  I  dv.  From  (134)  e  =  kp}  or  the 
elasticity  in  adiabatic  transformations  is  k  times  as  great  as  in  isothermal 
transformations.  This  is  important  in  the  theory  of  sound,  since  it  is  the 
adiabatic  elasticity  which  determines  the  velocity  of  sound  waves. 

The  ratio  k  is  nearly  the  same  for  all  the  permanent  gases  and  equals 
1.4  (see  section  135).  It  may  be  determined  experimentally  by  the 
method  of  Jamin  and  Richartz.  A  calibrated  glass  vessel  communicating 
with  a  mercury  manometer  is  heated  by  a  wire  carrying  an  electric  current 
and  filled  with  the  gas.  If  the  pressure  be  kept  constant  for  a  time  /, 

.   (135)  H^»(7-  T^t- 


ORIGIN    AND    MAINTENANCE    OF    SUN'S    HEAT.  95 

If  the  volume  be  kept  constant  and  the  heat  applied  for  an  equal  time, 

From  which,  assuming  H^  =  Hz, 

(137)  k=     2  _     °  =  -,    2_    °     °. 

The  pressures   are  determined  by  the  manometer,  and  the  volumes 
from  the  calibration  of  the  vessel. 

186.  Origin  and  Maintenance  of  Sun's  Heat. — There  has  been 
no  great  change  in  the  amount  of  heat  received  by  the  earth  from  the  sun 
in  thousands  of  years.     Langley  has  shown  that  the  earth  receives  energy 
from  the  sun  at  the  rate  of  about  2.5  horse-power  per  square  meter,  or 
the  latter  must  radiate  from  its  surface  more  than  100,000  horse-power  per 
square  meter.     If  the  sun  were  solid  coal  it  could  not  maintain  this  supply 
by  combustion  for  3000   years.       There  are  two    dynamical  theories  to 
account  for  this  heat:  Mayer's  meteoric  theory  is  that  the  heat  is  main- 
tained by  the  fall  of  meteors  on  the  sun.     The  earth,  if  it  fell  into  the 
sun,  would  generate  as  much  heat  as  would  be  furnished  by  the  combus- 
tion of  5600  globes  of  coal.     Nevertheless  this  theory  is  hardly  sufficient. 
Helmholtz's  contraction  theory  is  that  the  heat  is  evolved  by  the  con- 
traction of  the  sun,  considered  as  a  partly  gaseous  body.      From  thermo- 
dynamics it  may  be  shown  that  a  contraction  of  diameter  of  250  meters 
per  year  would  maintain  a  constant  temperature,  and  such  a  shrinkage 
would  not  be  observed  for  ten  thousand  years.     When  the  sun  is  mostly 
liquefied  or  solidified  this  action  will  cease  and  it  will  grow  cooler.     The 
interior  heat  of  the  earth  was  probably  originally  produced  by  a  similar 
process,  which  has  now  ceased,  and  it  is  constantly  being  lost  by  con- 
duction to  the  surface  and  radiation  into  space. 

187.  Dissipation  and  Degradation  of  Energy. — From  the  effi- 
ciency  equation    (128),    it  is   evident   that   only   a   fraction  of  a  given 
quantity  of  heat  can  ever  be  converted  into  work.     The  remainder  is  lost 
by  conduction  or  radiation  to  cooler  bodies.     There  is  thus  a  constant 
tendency  for  the  universe  to  come  to  a  uniform  temperature.     This  has 
been  called  by  Kelvin  the  dissipation  or  degradation  of  energy.     When 
universal-temperature  equilibrium  exists  there  can  be  no  available  energy, 
unless  by  some  process  (Maxwell's   "demon")  the  individual  molecules 
possessing  greater  kinetic  energy  than  the  average  can  be  sorted  from 
those  having  less. 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 
LOAN  DEPT. 

This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 
Renewed  books  are  subject  to  immediate  recall. 


prrrt'n  !  n 

*'               *    *n—  -> 

**urt     4  V£b3 

. 

General  Library 
LD  21A-50m-8,'57                                University  of  California 
(C8481slO)476B                                                 Berkeley 

VC   I  1418 


